This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages) This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (December 2014) (Learn how and when to remove this template message) | This article may be too long to read and navigate comfortably. The readable prose size is 409 kilobytes. Please consider splitting content into sub-articles, condensing it, or adding subheadings. Please discuss this issue on the article's talk page. (December 2014) | (Learn how and when to remove this template message) In the mathematical field of group theory, an Artin transfer is a certain homomorphism from an arbitrary finite or infinite group to the commutator quotient group of a subgroup of finite index. Originally, such mappings arose as group theoretic counterparts of class extension homomorphisms of abelian extensions of algebraic number fields by applying Artin's reciprocity maps to ideal class groups and analyzing the resulting homomorphisms between quotients of Galois groups. However, independently of number theoretic applications, a partial order on the kernels and targets of Artin transfers has recently turned out to be compatible with parent-descendant relations between finite p-groups (with a prime number p), which can be visualized in descendant trees. Therefore, Artin transfers provide a valuable tool for the classification of finite p-groups and for searching and identifying particular groups in descendant trees by looking for patterns defined by the kernels and targets of Artin transfers. These strategies of pattern recognition are useful in purely group theoretic context, as well as for applications in algebraic number theory concerning Galois groups of higher p-class fields and Hilbert p-class field towers. ## Contents * 1 Transversals of a subgroup * 2 Permutation representation * 3 Artin transfer * 3.1 Independence of the transversal * 3.2 Artin transfers as homomorphisms * 3.3 Wreath product of H and S(n) * 3.4 Composition of Artin transfers * 3.5 Wreath product of S(m) and S(n) * 3.6 Cycle decomposition * 3.7 Transfer to a normal subgroup * 4 Computational implementation * 4.1 Abelianization of type (p,p) * 4.2 Abelianization of type (p2,p) * 5 Transfer kernels and targets * 6 Abelianization of type (p,p) * 7 Abelianization of type (p2,p) * 7.1 First layer * 7.2 Second layer * 7.3 Transfer kernel type * 7.4 Connections between layers * 8 Inheritance from quotients * 8.1 Passing through the abelianization * 8.2 TTT singulets * 8.3 TKT singulets * 8.4 TTT and TKT multiplets * 8.5 Inherited automorphisms * 9 Stabilization criteria * 10 Structured descendant trees (SDTs) * 11 Pattern recognition * 11.1 Historical example * 12 Commutator calculus * 13 Systematic library of SDTs * 13.1 Coclass 1 * 13.2 Coclass 2 * 13.2.1 Abelianization of type (p,p) * 13.2.2 Abelianization of type (p2,p) * 13.2.3 Abelianization of type (p,p,p) * 13.3 Coclass 3 * 13.3.1 Abelianization of type (p2,p) * 13.3.2 Abelianization of type (p,p,p) * 14 Arithmetical applications * 14.1 Example * 14.2 Comparison of various primes * 15 References ## Transversals of a subgroup[edit] Let G {\displaystyle G} be a group and H ≤ G {\displaystyle H\leq G} be a subgroup of finite index n . {\displaystyle n.} Definitions.[1] A left transversal of H {\displaystyle H} in G {\displaystyle G} is an ordered system ( g 1 , … , g n ) {\displaystyle (g_{1},\ldots ,g_{n})} of representatives for the left cosets of H {\displaystyle H} in G {\displaystyle G} such that G = ⨆ i = 1 n g i H . {\displaystyle G=\bigsqcup _{i=1}^{n}g_{i}H.} Similarly a right transversal of H {\displaystyle H} in G {\displaystyle G} is an ordered system ( d 1 , … , d n ) {\displaystyle (d_{1},\ldots ,d_{n})} of representatives for the right cosets of H {\displaystyle H} in G {\displaystyle G} such that G = ⨆ i = 1 n H d i . {\displaystyle G=\bigsqcup _{i=1}^{n}Hd_{i}.} Remark. For any transversal of H {\displaystyle H} in G {\displaystyle G} , there exists a unique subscript 1 ≤ i 0 ≤ n {\displaystyle 1\leq i_{0}\leq n} such that g i 0 ∈ H {\displaystyle g_{i_{0}}\in H} , resp. d i 0 ∈ H {\displaystyle d_{i_{0}}\in H} . Of course, this element with subscript i 0 {\displaystyle i_{0}} which represents the principal coset (i.e., the subgroup H {\displaystyle H} itself) may be, but need not be, replaced by the neutral element 1 {\displaystyle 1} . Lemma.[2] Let G {\displaystyle G} be a non-abelian group with subgroup H {\displaystyle H} . Then the inverse elements ( g 1 − 1 , … , g n − 1 ) {\displaystyle (g_{1}^{-1},\ldots ,g_{n}^{-1})} of a left transversal ( g 1 , … , g n ) {\displaystyle (g_{1},\ldots ,g_{n})} of H {\displaystyle H} in G {\displaystyle G} form a right transversal of H {\displaystyle H} in G {\displaystyle G} . Moreover, if H {\displaystyle H} is a normal subgroup of G {\displaystyle G} , then any left transversal is also a right transversal of H {\displaystyle H} in G {\displaystyle G} . Proof. Since the mapping x ↦ x − 1 {\displaystyle x\mapsto x^{-1}} is an involution of G {\displaystyle G} we see that: G = G − 1 = ⨆ i = 1 n ( g i H ) − 1 = ⨆ i = 1 n H − 1 g i − 1 = ⨆ i = 1 n H g i − 1 . {\displaystyle G=G^{-1}=\bigsqcup _{i=1}^{n}(g_{i}H)^{-1}=\bigsqcup _{i=1}^{n}H^{-1}g_{i}^{-1}=\bigsqcup _{i=1}^{n}Hg_{i}^{-1}.} For a normal subgroup H {\displaystyle H} we have x H = H x {\displaystyle xH=Hx} for each x ∈ G {\displaystyle x\in G} . We must check when the image of a transversal under a homomorphism is also a transversal. Proposition. Let ϕ : G -> K {\displaystyle \phi :G\to K} be a group homomorphism and ( g 1 , … , g n ) {\displaystyle (g_{1},\ldots ,g_{n})} be a left transversal of a subgroup H {\displaystyle H} in G {\displaystyle G} with finite index n . {\displaystyle n.} The following two conditions are equivalent: * ( ϕ ( g 1 ) , … , ϕ ( g n ) ) {\displaystyle (\phi (g_{1}),\ldots ,\phi (g_{n}))} is a left transversal of the subgroup ϕ ( H ) {\displaystyle \phi (H)} in the image ϕ ( G ) {\displaystyle \phi (G)} with finite index ( ϕ ( G ) : ϕ ( H ) ) = n . {\displaystyle (\phi (G):\phi (H))=n.} * ker ⁡ ( ϕ ) ≤ H . {\displaystyle \ker(\phi )\leq H.} Proof. As a mapping of sets ϕ {\displaystyle \phi } maps the union to another union: ϕ ( G ) = ϕ ( ⋃ i = 1 n g i H ) = ⋃ i = 1 n ϕ ( g i H ) = ⋃ i = 1 n ϕ ( g i ) ϕ ( H ) , {\displaystyle \phi (G)=\phi \left(\bigcup _{i=1}^{n}g_{i}H\right)=\bigcup _{i=1}^{n}\phi (g_{i}H)=\bigcup _{i=1}^{n}\phi (g_{i})\phi (H),} but weakens the equality for the intersection to a trivial inclusion: ∅ = ϕ ( ∅ ) = ϕ ( g i H ∩ g j H ) ⊆ ϕ ( g i H ) ∩ ϕ ( g j H ) = ϕ ( g i ) ϕ ( H ) ∩ ϕ ( g j ) ϕ ( H ) , i ≠ j . {\displaystyle \emptyset =\phi (\emptyset )=\phi (g_{i}H\cap g_{j}H)\subseteq \phi (g_{i}H)\cap \phi (g_{j}H)=\phi (g_{i})\phi (H)\cap \phi (g_{j})\phi (H),\qquad i\neq j.} Suppose for some 1 ≤ i ≤ j ≤ n {\displaystyle 1\leq i\leq j\leq n} : ϕ ( g i ) ϕ ( H ) ∩ ϕ ( g j ) ϕ ( H ) ≠ ∅ {\displaystyle \phi (g_{i})\phi (H)\cap \phi (g_{j})\phi (H)\neq \emptyset } then there exists elements h i , h j ∈ H {\displaystyle h_{i},h_{j}\in H} such that ϕ ( g i ) ϕ ( h i ) = ϕ ( g j ) ϕ ( h j ) {\displaystyle \phi (g_{i})\phi (h_{i})=\phi (g_{j})\phi (h_{j})} Then we have: ϕ ( g i ) ϕ ( h i ) = ϕ ( g j ) ϕ ( h j ) ⟹ ϕ ( g j ) − 1 ϕ ( g i ) ϕ ( h i ) ϕ ( h j ) − 1 = 1 ⟹ ϕ ( g j − 1 g i h i h j − 1 ) = 1 ⟹ g j − 1 g i h i h j − 1 ∈ ker ⁡ ( ϕ ) ⟹ g j − 1 g i h i h j − 1 ∈ H ker ⁡ ( ϕ ) ≤ H ⟹ g j − 1 g i ∈ H h i h j − 1 ∈ H ⟹ g i H = g j H ⟹ i = j {\displaystyle {\begin{aligned}\phi (g_{i})\phi (h_{i})=\phi (g_{j})\phi (h_{j})&\Longrightarrow \phi (g_{j})^{-1}\phi (g_{i})\phi (h_{i})\phi (h_{j})^{-1}=1\\\&\Longrightarrow \phi \left(g_{j}^{-1}g_{i}h_{i}h_{j}^{-1}\right)=1\\\&\Longrightarrow g_{j}^{-1}g_{i}h_{i}h_{j}^{-1}\in \ker(\phi )\\\&\Longrightarrow g_{j}^{-1}g_{i}h_{i}h_{j}^{-1}\in H&&\ker(\phi )\leq H\\\&\Longrightarrow g_{j}^{-1}g_{i}\in H&&h_{i}h_{j}^{-1}\in H\\\&\Longrightarrow g_{i}H=g_{j}H\\\&\Longrightarrow i=j\end{aligned}}} Conversely if ker ⁡ ( ϕ ) ⊈ H {\displaystyle \ker(\phi )\nsubseteq H} then there exists x ∈ G ∖ H {\displaystyle x\in G\setminus H} such that ϕ ( x ) = 1. {\displaystyle \phi (x)=1.} But the homomorphism ϕ {\displaystyle \phi } maps the disjoint cosets x ⋅ H ∩ 1 ⋅ H = ∅ {\displaystyle x\cdot H\cap 1\cdot H=\emptyset } to equal cosets: ϕ ( x ) ϕ ( H ) ∩ ϕ ( 1 ) ϕ ( H ) = 1 ⋅ ϕ ( H ) ∩ 1 ⋅ ϕ ( H ) = ϕ ( H ) . {\displaystyle \phi (x)\phi (H)\cap \phi (1)\phi (H)=1\cdot \phi (H)\cap 1\cdot \phi (H)=\phi (H).} Remark. We emphasize the important equivalence of the proposition in a formula: ( 1 ) ker ⁡ ( ϕ ) ≤ H ⟺ { ϕ ( G ) = ⨆ i = 1 n ϕ ( g i ) ϕ ( H ) ( ϕ ( G ) : ϕ ( H ) ) = n {\displaystyle (1)\quad \ker(\phi )\leq H\quad \Longleftrightarrow \quad {\begin{cases}\phi (G)=\bigsqcup _{i=1}^{n}\phi (g_{i})\phi (H)\\\\(\phi (G):\phi (H))=n\end{cases}}} ## Permutation representation[edit] Suppose ( g 1 , … , g n ) {\displaystyle (g_{1},\ldots ,g_{n})} is a left transversal of a subgroup H {\displaystyle H} of finite index n {\displaystyle n} in a group G {\displaystyle G} . A fixed element x ∈ G {\displaystyle x\in G} gives rise to a unique permutation π x ∈ S n {\displaystyle \pi _{x}\in S_{n}} of the left cosets of H {\displaystyle H} in G {\displaystyle G} by left multiplication such that: ( 2 ) ∀ i ∈ { 1 , … , n } : x g i H = g π x ( i ) H ⟹ x g i ∈ g π x ( i ) H . {\displaystyle (2)\quad \forall i\in \\{1,\ldots ,n\\}:\qquad xg_{i}H=g_{\pi _{x}(i)}H\Longrightarrow xg_{i}\in g_{\pi _{x}(i)}H.} Using this we define a set of elements called the monomials associated with x {\displaystyle x} with respect to ( g 1 , … , g n ) {\displaystyle (g_{1},\ldots ,g_{n})} : ∀ i ∈ { 1 , … , n } : u x ( i ) := g π x ( i ) − 1 x g i ∈ H . {\displaystyle \forall i\in \\{1,\ldots ,n\\}:\qquad u_{x}(i):=g_{\pi _{x}(i)}^{-1}xg_{i}\in H.} Similarly, if ( d 1 , … , d n ) {\displaystyle (d_{1},\ldots ,d_{n})} is a right transversal of H {\displaystyle H} in G {\displaystyle G} , then a fixed element x ∈ G {\displaystyle x\in G} gives rise to a unique permutation ρ x ∈ S n {\displaystyle \rho _{x}\in S_{n}} of the right cosets of H {\displaystyle H} in G {\displaystyle G} by right multiplication such that: ( 3 ) ∀ i ∈ { 1 , … , n } : H d i x = H d ρ x ( i ) ⟹ d i x ∈ H d ρ x ( i ) . {\displaystyle (3)\quad \forall i\in \\{1,\ldots ,n\\}:\qquad Hd_{i}x=Hd_{\rho _{x}(i)}\Longrightarrow d_{i}x\in Hd_{\rho _{x}(i)}.} And we define the monomials associated with x {\displaystyle x} with respect to ( d 1 , … , d n ) {\displaystyle (d_{1},\ldots ,d_{n})} : ∀ i ∈ { 1 , … , n } : w x ( i ) := d i x d ρ x ( i ) − 1 ∈ H . {\displaystyle \forall i\in \\{1,\ldots ,n\\}:\qquad w_{x}(i):=d_{i}xd_{\rho _{x}(i)}^{-1}\in H.} Definition.[1] The mappings: { G -> S n x ↦ π x { G -> S n x ↦ ρ x {\displaystyle {\begin{cases}G\to S_{n}\\\x\mapsto \pi _{x}\end{cases}}\qquad {\begin{cases}G\to S_{n}\\\x\mapsto \rho _{x}\end{cases}}} are called the permutation representation of G {\displaystyle G} in the symmetric group S n {\displaystyle S_{n}} with respect to ( g 1 , … , g n ) {\displaystyle (g_{1},\ldots ,g_{n})} and ( d 1 , … , d n ) {\displaystyle (d_{1},\ldots ,d_{n})} respectively. Definition.[1] The mappings: { G -> H n × S n x ↦ ( u x ( 1 ) , … , u x ( n ) ; π x ) { G -> H n × S n x ↦ ( w x ( 1 ) , … , w x ( n ) ; ρ x ) {\displaystyle {\begin{cases}G\to H^{n}\times S_{n}\\\x\mapsto (u_{x}(1),\ldots ,u_{x}(n);\pi _{x})\end{cases}}\qquad {\begin{cases}G\to H^{n}\times S_{n}\\\x\mapsto (w_{x}(1),\ldots ,w_{x}(n);\rho _{x})\end{cases}}} are called the monomial representation of G {\displaystyle G} in H n × S n {\displaystyle H^{n}\times S_{n}} with respect to ( g 1 , … , g n ) {\displaystyle (g_{1},\ldots ,g_{n})} and ( d 1 , … , d n ) {\displaystyle (d_{1},\ldots ,d_{n})} respectively. Lemma. For the right transversal ( g 1 − 1 , … , g n − 1 ) {\displaystyle (g_{1}^{-1},\ldots ,g_{n}^{-1})} associated to the left transversal ( g 1 , … , g n ) {\displaystyle (g_{1},\ldots ,g_{n})} , we have the following relations between the monomials and permutations corresponding to an element x ∈ G {\displaystyle x\in G} : ( 4 ) { w x − 1 ( i ) = u x ( i ) − 1 1 ≤ i ≤ n ρ x − 1 = π x {\displaystyle (4)\quad {\begin{cases}w_{x^{-1}}(i)=u_{x}(i)^{-1}&1\leq i\leq n\\\\\rho _{x^{-1}}=\pi _{x}\end{cases}}} Proof. For the right transversal ( g 1 − 1 , … , g n − 1 ) {\displaystyle (g_{1}^{-1},\ldots ,g_{n}^{-1})} , we have w x ( i ) = g i − 1 x g ρ x ( i ) {\displaystyle w_{x}(i)=g_{i}^{-1}xg_{\rho _{x}(i)}} , for each 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} . On the other hand, for the left transversal ( g 1 , … , g n ) {\displaystyle (g_{1},\ldots ,g_{n})} , we have ∀ i ∈ { 1 , … , n } : u x ( i ) − 1 = ( g π x ( i ) − 1 x g i ) − 1 = g i − 1 x − 1 g π x ( i ) = g i − 1 x − 1 g ρ x − 1 ( i ) = w x − 1 ( i ) . {\displaystyle \forall i\in \\{1,\ldots ,n\\}:\qquad u_{x}(i)^{-1}=\left(g_{\pi _{x}(i)}^{-1}xg_{i}\right)^{-1}=g_{i}^{-1}x^{-1}g_{\pi _{x}(i)}=g_{i}^{-1}x^{-1}g_{\rho _{x^{-1}}(i)}=w_{x^{-1}}(i).} This relation simultaneously shows that, for any x ∈ G {\displaystyle x\in G} , the permutation representations and the associated monomials are connected by ρ x − 1 = π x {\displaystyle \rho _{x^{-1}}=\pi _{x}} and w x − 1 ( i ) = u x ( i ) − 1 {\displaystyle w_{x^{-1}}(i)=u_{x}(i)^{-1}} for each 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} . ## Artin transfer[edit] Definitions.[2][3] Let G {\displaystyle G} be a group and H {\displaystyle H} a subgroup of finite index n . {\displaystyle n.} Assume ( g ) = ( g 1 , … , g n ) {\displaystyle (g)=(g_{1},\ldots ,g_{n})} is a left transversal of H {\displaystyle H} in G {\displaystyle G} with associated permutation representation π x : G -> S n , {\displaystyle \pi _{x}:G\to S_{n},} such that ∀ i ∈ { 1 , … , n } : u x ( i ) := g π x ( i ) − 1 x g i ∈ H . {\displaystyle \forall i\in \\{1,\ldots ,n\\}:\qquad u_{x}(i):=g_{\pi _{x}(i)}^{-1}xg_{i}\in H.} Similarly let ( d ) = ( d 1 , … , d n ) {\displaystyle (d)=(d_{1},\ldots ,d_{n})} be a right transversal of H {\displaystyle H} in G {\displaystyle G} with associated permutation representation ρ x : G -> S n {\displaystyle \rho _{x}:G\to S_{n}} such that ∀ i ∈ { 1 , … , n } : w x ( i ) := d i x d ρ x ( i ) − 1 ∈ H . {\displaystyle \forall i\in \\{1,\ldots ,n\\}:\qquad w_{x}(i):=d_{i}xd_{\rho _{x}(i)}^{-1}\in H.} The Artin transfer T G , H ( g ) : G -> H / H ′ {\displaystyle T_{G,H}^{(g)}:G\to H/H'} with respect to ( g 1 , … , g n ) {\displaystyle (g_{1},\ldots ,g_{n})} is defined as: ( 5 ) ∀ x ∈ G : T G , H ( g ) ( x ) := ∏ i = 1 n g π x ( i ) − 1 x g i ⋅ H ′ = ∏ i = 1 n u x ( i ) ⋅ H ′ . {\displaystyle (5)\quad \forall x\in G:\qquad T_{G,H}^{(g)}(x):=\prod _{i=1}^{n}g_{\pi _{x}(i)}^{-1}xg_{i}\cdot H'=\prod _{i=1}^{n}u_{x}(i)\cdot H'.} Similarly we define: ( 6 ) ∀ x ∈ G : T G , H ( d ) ( x ) := ∏ i = 1 n d i x d ρ x ( i ) − 1 ⋅ H ′ = ∏ i = 1 n w x ( i ) ⋅ H ′ . {\displaystyle (6)\quad \forall x\in G:\qquad T_{G,H}^{(d)}(x):=\prod _{i=1}^{n}d_{i}xd_{\rho _{x}(i)}^{-1}\cdot H'=\prod _{i=1}^{n}w_{x}(i)\cdot H'.} Remarks. Isaacs[4] calls the mappings { P : G -> H x ↦ ∏ i = 1 n u x ( i ) { P : G -> H x ↦ ∏ i = 1 n w x ( i ) {\displaystyle {\begin{cases}P:G\to H\\\x\mapsto \prod _{i=1}^{n}u_{x}(i)\end{cases}}\qquad {\begin{cases}P:G\to H\\\x\mapsto \prod _{i=1}^{n}w_{x}(i)\end{cases}}} the pre-transfer from G {\displaystyle G} to H {\displaystyle H} . The pre-transfer can be composed with a homomorphism ϕ : H -> A {\displaystyle \phi :H\to A} from H {\displaystyle H} into an abelian group A {\displaystyle A} to define a more general version of the transfer from G {\displaystyle G} to A {\displaystyle A} via ϕ {\displaystyle \phi } , which occurs in the book by Gorenstein.[5] { ( ϕ ∘ P ) : G -> A x ↦ ∏ i = 1 n ϕ ( u x ( i ) ) { ( ϕ ∘ P ) : G -> A x ↦ ∏ i = 1 n ϕ ( w x ( i ) ) {\displaystyle {\begin{cases}(\phi \circ P):G\to A\\\x\mapsto \prod _{i=1}^{n}\phi (u_{x}(i))\end{cases}}\qquad {\begin{cases}(\phi \circ P):G\to A\\\x\mapsto \prod _{i=1}^{n}\phi (w_{x}(i))\end{cases}}} Taking the natural epimorphism { ϕ : H -> H / H ′ v ↦ v H ′ {\displaystyle {\begin{cases}\phi :H\to H/H'\\\v\mapsto vH'\end{cases}}} yields the preceding definition of the Artin transfer T G , H {\displaystyle T_{G,H}} in its original form by Schur[2] and by Emil Artin,[3] which has also been dubbed Verlagerung by Hasse.[6] Note that, in general, the pre-transfer is neither independent of the transversal nor a group homomorphism. ### Independence of the transversal[edit] Proposition.[1][2][4][5][7][8][9] The Artin transfers with respect to any two left transversals of H {\displaystyle H} in G {\displaystyle G} coincide. Proof. Let ( ℓ ) = ( ℓ 1 , … , ℓ n ) {\displaystyle (\ell )=(\ell _{1},\ldots ,\ell _{n})} and ( g ) = ( g 1 , … , g n ) {\displaystyle (g)=(g_{1},\ldots ,g_{n})} be two left transversals of H {\displaystyle H} in G {\displaystyle G} . Then there exists a unique permutation σ ∈ S n {\displaystyle \sigma \in S_{n}} such that: ∀ i ∈ { 1 , … , n } : g i H = ℓ σ ( i ) H . {\displaystyle \forall i\in \\{1,\ldots ,n\\}:\qquad g_{i}H=\ell _{\sigma (i)}H.} Consequently: ∀ i ∈ { 1 , … , n } , ∃ h i ∈ H : g i h i = ℓ σ ( i ) . {\displaystyle \forall i\in \\{1,\ldots ,n\\},\exists h_{i}\in H:\qquad g_{i}h_{i}=\ell _{\sigma (i)}.} For a fixed element x ∈ G {\displaystyle x\in G} , there exists a unique permutation λ x ∈ S n {\displaystyle \lambda _{x}\in S_{n}} such that: ∀ i ∈ { 1 , … , n } : ℓ λ x ( σ ( i ) ) H = x ℓ σ ( i ) H = x g i h i H = x g i H = g π x ( i ) H = g π x ( i ) h π x ( i ) H = ℓ σ ( π x ( i ) ) H . {\displaystyle \forall i\in \\{1,\ldots ,n\\}:\qquad \ell _{\lambda _{x}(\sigma (i))}H=x\ell _{\sigma (i)}H=xg_{i}h_{i}H=xg_{i}H=g_{\pi _{x}(i)}H=g_{\pi _{x}(i)}h_{\pi _{x}(i)}H=\ell _{\sigma (\pi _{x}(i))}H.} Therefore, the permutation representation of G {\displaystyle G} with respect to ( ℓ 1 , … , ℓ n ) {\displaystyle (\ell _{1},\ldots ,\ell _{n})} is given by λ x ∘ σ = σ ∘ π x {\displaystyle \lambda _{x}\circ \sigma =\sigma \circ \pi _{x}} which yields: λ x = σ ∘ π x ∘ σ − 1 ∈ S n . {\displaystyle \lambda _{x}=\sigma \circ \pi _{x}\circ \sigma ^{-1}\in S_{n}.} Furthermore, for the connection between the two elements: v x ( i ) := ℓ λ x ( i ) − 1 x ℓ i ∈ H u x ( i ) := g π x ( i ) − 1 x g i ∈ H {\displaystyle {\begin{aligned}v_{x}(i)&:=\ell _{\lambda _{x}(i)}^{-1}x\ell _{i}\in H\\\u_{x}(i)&:=g_{\pi _{x}(i)}^{-1}xg_{i}\in H\end{aligned}}} we have: ∀ i ∈ { 1 , … , n } : v x ( σ ( i ) ) = ℓ λ x ( σ ( i ) ) − 1 x ℓ σ ( i ) = ℓ σ ( π x ( i ) ) − 1 x g i h i = ( g π x ( i ) h π x ( i ) ) − 1 x g i h i = h π x ( i ) − 1 g π x ( i ) − 1 x g i h i = h π x ( i ) − 1 u x ( i ) h i . {\displaystyle \forall i\in \\{1,\ldots ,n\\}:\qquad v_{x}(\sigma (i))=\ell _{\lambda _{x}(\sigma (i))}^{-1}x\ell _{\sigma (i)}=\ell _{\sigma (\pi _{x}(i))}^{-1}xg_{i}h_{i}=\left(g_{\pi _{x}(i)}h_{\pi _{x}(i)}\right)^{-1}xg_{i}h_{i}=h_{\pi _{x}(i)}^{-1}g_{\pi _{x}(i)}^{-1}xg_{i}h_{i}=h_{\pi _{x}(i)}^{-1}u_{x}(i)h_{i}.} Finally since H / H ′ {\displaystyle H/H'} is abelian and σ {\displaystyle \sigma } and π x {\displaystyle \pi _{x}} are permutations, the Artin transfer turns out to be independent of the left transversal: T G , H ( ℓ ) ( x ) = ∏ i = 1 n v x ( σ ( i ) ) ⋅ H ′ = ∏ i = 1 n h π x ( i ) − 1 u x ( i ) h i ⋅ H ′ = ∏ i = 1 n u x ( i ) ∏ i = 1 n h π x ( i ) − 1 ∏ i = 1 n h i ⋅ H ′ = ∏ i = 1 n u x ( i ) ⋅ 1 ⋅ H ′ = ∏ i = 1 n u x ( i ) ⋅ H ′ = T G , H ( g ) ( x ) , {\displaystyle T_{G,H}^{(\ell )}(x)=\prod _{i=1}^{n}v_{x}(\sigma (i))\cdot H'=\prod _{i=1}^{n}h_{\pi _{x}(i)}^{-1}u_{x}(i)h_{i}\cdot H'=\prod _{i=1}^{n}u_{x}(i)\prod _{i=1}^{n}h_{\pi _{x}(i)}^{-1}\prod _{i=1}^{n}h_{i}\cdot H'=\prod _{i=1}^{n}u_{x}(i)\cdot 1\cdot H'=\prod _{i=1}^{n}u_{x}(i)\cdot H'=T_{G,H}^{(g)}(x),} as defined in formula (5). Proposition. The Artin transfers with respect to any two right transversals of H {\displaystyle H} in G {\displaystyle G} coincide. Proof. Similar to the previous proposition. Proposition. The Artin transfers with respect to ( g − 1 ) = ( g 1 − 1 , … , g n − 1 ) {\displaystyle (g^{-1})=(g_{1}^{-1},\ldots ,g_{n}^{-1})} and ( g ) = ( g 1 , … , g n ) {\displaystyle (g)=(g_{1},\ldots ,g_{n})} coincide. Proof. Using formula (4) and H / H ′ {\displaystyle H/H'} being abelian we have: T G , H ( g − 1 ) ( x ) = ∏ i = 1 n g i − 1 x g ρ x ( i ) ⋅ H ′ = ∏ i = 1 n w x ( i ) ⋅ H ′ = ∏ i = 1 n u x − 1 ( i ) − 1 ⋅ H ′ = ( ∏ i = 1 n u x − 1 ( i ) ⋅ H ′ ) − 1 = ( T G , H ( g ) ( x − 1 ) ) − 1 = T G , H ( g ) ( x ) . {\displaystyle T_{G,H}^{(g^{-1})}(x)=\prod _{i=1}^{n}g_{i}^{-1}xg_{\rho _{x}(i)}\cdot H'=\prod _{i=1}^{n}w_{x}(i)\cdot H'=\prod _{i=1}^{n}u_{x^{-1}}(i)^{-1}\cdot H'=\left(\prod _{i=1}^{n}u_{x^{-1}}(i)\cdot H'\right)^{-1}=\left(T_{G,H}^{(g)}\left(x^{-1}\right)\right)^{-1}=T_{G,H}^{(g)}(x).} The last step is justified by the fact that the Artin transfer is a homomorphism. This will be shown in the following section. Corollary. The Artin transfer is independent of the choice of transversals and only depends on H {\displaystyle H} and G {\displaystyle G} . ### Artin transfers as homomorphisms[edit] Theorem.[1][2][4][5][7][8][9] Let ( g 1 , … , g n ) {\displaystyle (g_{1},\ldots ,g_{n})} be a left transversal of H {\displaystyle H} in G {\displaystyle G} . The Artin transfer { T G , H : G -> H / H ′ x ↦ ∏ i = 1 n g π x ( i ) − 1 x g i ⋅ H ′ {\displaystyle {\begin{cases}T_{G,H}:G\to H/H'\\\x\mapsto \prod _{i=1}^{n}g_{\pi _{x}(i)}^{-1}xg_{i}\cdot H'\end{cases}}} and the permutation representation: { G -> S n x ↦ π x {\displaystyle {\begin{cases}G\to S_{n}\\\x\mapsto \pi _{x}\end{cases}}} are group homomorphisms: ( 7 ) ∀ x , y ∈ G : T G , H ( x y ) = T G , H ( x ) ⋅ T G , H ( y ) and π x y = π x ∘ π y . {\displaystyle (7)\quad \forall x,y\in G:\qquad T_{G,H}(xy)=T_{G,H}(x)\cdot T_{G,H}(y)\quad {\text{and}}\quad \pi _{xy}=\pi _{x}\circ \pi _{y}.} Proof Let x , y ∈ G {\displaystyle x,y\in G} : T G , H ( x ) ⋅ T G , H ( y ) = ∏ i = 1 n g π x ( i ) − 1 x g i H ′ ⋅ ∏ j = 1 n g π y ( j ) − 1 y g j ⋅ H ′ {\displaystyle T_{G,H}(x)\cdot T_{G,H}(y)=\prod _{i=1}^{n}g_{\pi _{x}(i)}^{-1}xg_{i}H'\cdot \prod _{j=1}^{n}g_{\pi _{y}(j)}^{-1}yg_{j}\cdot H'} Since H / H ′ {\displaystyle H/H'} is abelian and π y {\displaystyle \pi _{y}} is a permutation, we can change the order of the factors in the product: ∏ i = 1 n g π x ( i ) − 1 x g i H ′ ⋅ ∏ j = 1 n g π y ( j ) − 1 y g j ⋅ H ′ = ∏ j = 1 n g π x ( π y ( j ) ) − 1 x g π y ( j ) H ′ ⋅ ∏ j = 1 n g π y ( j ) − 1 y g j ⋅ H ′ = ∏ j = 1 n g π x ( π y ( j ) ) − 1 x g π y ( j ) g π y ( j ) − 1 y g j ⋅ H ′ = ∏ j = 1 n g ( π x ∘ π y ) ( j ) ) − 1 x y g j ⋅ H ′ = T G , H ( x y ) {\displaystyle {\begin{aligned}\prod _{i=1}^{n}g_{\pi _{x}(i)}^{-1}xg_{i}H'\cdot \prod _{j=1}^{n}g_{\pi _{y}(j)}^{-1}yg_{j}\cdot H'&=\prod _{j=1}^{n}g_{\pi _{x}(\pi _{y}(j))}^{-1}xg_{\pi _{y}(j)}H'\cdot \prod _{j=1}^{n}g_{\pi _{y}(j)}^{-1}yg_{j}\cdot H'\\\&=\prod _{j=1}^{n}g_{\pi _{x}(\pi _{y}(j))}^{-1}xg_{\pi _{y}(j)}g_{\pi _{y}(j)}^{-1}yg_{j}\cdot H'\\\&=\prod _{j=1}^{n}g_{(\pi _{x}\circ \pi _{y})(j))}^{-1}xyg_{j}\cdot H'\\\&=T_{G,H}(xy)\end{aligned}}} This relation simultaneously shows that the Artin transfer and the permutation representation are homomorphisms. It is illuminating to restate the homomorphism property of the Artin transfer in terms of the monomial representation. The images of the factors x , y {\displaystyle x,y} are given by T G , H ( x ) = ∏ i = 1 n u x ( i ) ⋅ H ′ and T G , H ( y ) = ∏ j = 1 n u y ( j ) ⋅ H ′ . {\displaystyle T_{G,H}(x)=\prod _{i=1}^{n}u_{x}(i)\cdot H'\quad {\text{and}}\quad T_{G,H}(y)=\prod _{j=1}^{n}u_{y}(j)\cdot H'.} In the last proof, the image of the product x y {\displaystyle xy} turned out to be T G , H ( x y ) = ∏ j = 1 n g π x ( π y ( j ) ) − 1 x g π y ( j ) g π y ( j ) − 1 y g j ⋅ H ′ = ∏ j = 1 n u x ( π y ( j ) ) ⋅ u y ( j ) ⋅ H ′ {\displaystyle T_{G,H}(xy)=\prod _{j=1}^{n}g_{\pi _{x}(\pi _{y}(j))}^{-1}xg_{\pi _{y}(j)}g_{\pi _{y}(j)}^{-1}yg_{j}\cdot H'=\prod _{j=1}^{n}u_{x}(\pi _{y}(j))\cdot u_{y}(j)\cdot H'} , which is a very peculiar law of composition discussed in more detail in the following section. The law is reminiscent of crossed homomorphisms x ↦ u x {\displaystyle x\mapsto u_{x}} in the first cohomology group H 1 ( G , M ) {\displaystyle \mathrm {H} ^{1}(G,M)} of a G {\displaystyle G} -module M {\displaystyle M} , which have the property u x y = u x y ⋅ u y {\displaystyle u_{xy}=u_{x}^{y}\cdot u_{y}} for x , y ∈ G {\displaystyle x,y\in G} . ### Wreath product of H and S(n)[edit] The peculiar structures which arose in the previous section can also be interpreted by endowing the cartesian product H n × S n {\displaystyle H^{n}\times S_{n}} with a special law of composition known as the wreath product H ≀ S n {\displaystyle H\wr S_{n}} of the groups H {\displaystyle H} and S n {\displaystyle S_{n}} with respect to the set { 1 , … , n } . {\displaystyle \\{1,\ldots ,n\\}.} Definition. For x , y ∈ G {\displaystyle x,y\in G} , the wreath product of the associated monomials and permutations is given by ( 8 ) ( u x ( 1 ) , … , u x ( n ) ; π x ) ⋅ ( u y ( 1 ) , … , u y ( n ) ; π y ) := ( u x ( π y ( 1 ) ) ⋅ u y ( 1 ) , … , u x ( π y ( n ) ) ⋅ u y ( n ) ; π x ∘ π y ) = ( u x y ( 1 ) , … , u x y ( n ) ; π x y ) . {\displaystyle (8)\quad (u_{x}(1),\ldots ,u_{x}(n);\pi _{x})\cdot (u_{y}(1),\ldots ,u_{y}(n);\pi _{y}):=(u_{x}(\pi _{y}(1))\cdot u_{y}(1),\ldots ,u_{x}(\pi _{y}(n))\cdot u_{y}(n);\pi _{x}\circ \pi _{y})=(u_{xy}(1),\ldots ,u_{xy}(n);\pi _{xy}).} Theorem.[1][7] With this law of composition on H n × S n {\displaystyle H^{n}\times S_{n}} the monomial representation { G -> H ≀ S n x ↦ ( u x ( 1 ) , … , u x ( n ) ; π x ) {\displaystyle {\begin{cases}G\to H\wr S_{n}\\\x\mapsto (u_{x}(1),\ldots ,u_{x}(n);\pi _{x})\end{cases}}} is an injective homomorphism. Proof The homomorphism property has been shown above already. For a homomorphism to be injective, it suffices to show the triviality of its kernel. The neutral element of the group H n × S n {\displaystyle H^{n}\times S_{n}} endowed with the wreath product is given by ( 1 , … , 1 ; 1 ) {\displaystyle (1,\ldots ,1;1)} , where the last 1 {\displaystyle 1} means the identity permutation. If ( u x ( 1 ) , … , u x ( n ) ; π x ) = ( 1 , … , 1 ; 1 ) {\displaystyle (u_{x}(1),\ldots ,u_{x}(n);\pi _{x})=(1,\ldots ,1;1)} , for some x ∈ G {\displaystyle x\in G} , then π x = 1 {\displaystyle \pi _{x}=1} and consequently ∀ i ∈ { 1 , … , n } : 1 = u x ( i ) = g π x ( i ) − 1 x g i = g i − 1 x g i . {\displaystyle \forall i\in \\{1,\ldots ,n\\}:\qquad 1=u_{x}(i)=g_{\pi _{x}(i)}^{-1}xg_{i}=g_{i}^{-1}xg_{i}.} Finally, an application of the inverse inner automorphism with g i {\displaystyle g_{i}} yields x = 1 {\displaystyle x=1} , as required for injectivity. Remark. The monomial representation of the theorem stands in contrast to the permutation representation, which cannot be injective if | G | > n ! . {\displaystyle |G|>n!.} Remark. Whereas Huppert[1] uses the monomial representation for defining the Artin transfer, we prefer to give the immediate definitions in formulas (5) and (6) and to merely illustrate the homomorphism property of the Artin transfer with the aid of the monomial representation. ### Composition of Artin transfers[edit] Theorem.[1][7] Let G {\displaystyle G} be a group with nested subgroups K ≤ H ≤ G {\displaystyle K\leq H\leq G} such that ( G : H ) = n , ( H : K ) = m {\displaystyle (G:H)=n,(H:K)=m} and ( G : K ) = ( G : H ) ⋅ ( H : K ) = n m < ∞ . {\displaystyle (G:K)=(G:H)\cdot (H:K)=nm<\infty .} Then the Artin transfer T G , K {\displaystyle T_{G,K}} is the compositum of the induced transfer T ~ H , K : H / H ′ -> K / K ′ {\displaystyle {\tilde {T}}_{H,K}:H/H'\to K/K'} and the Artin transfer T G , H {\displaystyle T_{G,H}} , that is: ( 9 ) T G , K = T ~ H , K ∘ T G , H {\displaystyle (9)\quad T_{G,K}={\tilde {T}}_{H,K}\circ T_{G,H}} . Proof If ( g 1 , … , g n ) {\displaystyle (g_{1},\ldots ,g_{n})} is a left transversal of H {\displaystyle H} in G {\displaystyle G} and ( h 1 , … , h m ) {\displaystyle (h_{1},\ldots ,h_{m})} is a left transversal of K {\displaystyle K} in H {\displaystyle H} , that is G = ⊔ i = 1 n g i H {\displaystyle G=\sqcup _{i=1}^{n}g_{i}H} and H = ⊔ j = 1 m h j K {\displaystyle H=\sqcup _{j=1}^{m}h_{j}K} , then G = ⨆ i = 1 n ⨆ j = 1 m g i h j K {\displaystyle G=\bigsqcup _{i=1}^{n}\bigsqcup _{j=1}^{m}g_{i}h_{j}K} is a disjoint left coset decomposition of G {\displaystyle G} with respect to K {\displaystyle K} . Given two elements x ∈ G {\displaystyle x\in G} and y ∈ H {\displaystyle y\in H} , there exist unique permutations π x ∈ S n {\displaystyle \pi _{x}\in S_{n}} , and σ y ∈ S m {\displaystyle \sigma _{y}\in S_{m}} , such that u x ( i ) := g π x ( i ) − 1 x g i ∈ H for all 1 ≤ i ≤ n v y ( j ) := h σ y ( j ) − 1 y h j ∈ K for all 1 ≤ j ≤ m {\displaystyle {\begin{aligned}u_{x}(i)&:=g_{\pi _{x}(i)}^{-1}xg_{i}\in H&&{\text{for all }}1\leq i\leq n\\\v_{y}(j)&:=h_{\sigma _{y}(j)}^{-1}yh_{j}\in K&&{\text{for all }}1\leq j\leq m\end{aligned}}} Then, anticipating the definition of the induced transfer, we have T G , H ( x ) = ∏ i = 1 n u x ( i ) ⋅ H ′ T ~ H , K ( y ⋅ H ′ ) = T H , K ( y ) = ∏ j = 1 m v y ( j ) ⋅ K ′ {\displaystyle {\begin{aligned}T_{G,H}(x)&=\prod _{i=1}^{n}u_{x}(i)\cdot H'\\\\{\tilde {T}}_{H,K}(y\cdot H')&=T_{H,K}(y)=\prod _{j=1}^{m}v_{y}(j)\cdot K'\end{aligned}}} For each pair of subscripts 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} and 1 ≤ j ≤ m {\displaystyle 1\leq j\leq m} , we put y i := u x ( i ) {\displaystyle y_{i}:=u_{x}(i)} , and we obtain x g i h j = g π x ( i ) g π x ( i ) − 1 x g i h j = g π x ( i ) u x ( i ) h j = g π x ( i ) y i h j = g π x ( i ) h σ y i ( j ) h σ y i ( j ) − 1 y i h j = g π x ( i ) h σ y i ( j ) v y i ( j ) , {\displaystyle xg_{i}h_{j}=g_{\pi _{x}(i)}g_{\pi _{x}(i)}^{-1}xg_{i}h_{j}=g_{\pi _{x}(i)}u_{x}(i)h_{j}=g_{\pi _{x}(i)}y_{i}h_{j}=g_{\pi _{x}(i)}h_{\sigma _{y_{i}}(j)}h_{\sigma _{y_{i}}(j)}^{-1}y_{i}h_{j}=g_{\pi _{x}(i)}h_{\sigma _{y_{i}}(j)}v_{y_{i}}(j),} resp. h σ y i ( j ) − 1 g π x ( i ) − 1 x g i h j = v y i ( j ) . {\displaystyle h_{\sigma _{y_{i}}(j)}^{-1}g_{\pi _{x}(i)}^{-1}xg_{i}h_{j}=v_{y_{i}}(j).} Therefore, the image of x {\displaystyle x} under the Artin transfer T G , K {\displaystyle T_{G,K}} is given by T G , K ( x ) = ∏ i = 1 n ∏ j = 1 m v y i ( j ) ⋅ K ′ = ∏ i = 1 n ∏ j = 1 m h σ y i ( j ) − 1 g π x ( i ) − 1 x g i h j ⋅ K ′ = ∏ i = 1 n ∏ j = 1 m h σ y i ( j ) − 1 u x ( i ) h j ⋅ K ′ = ∏ i = 1 n ∏ j = 1 m h σ y i ( j ) − 1 y i h j ⋅ K ′ = ∏ i = 1 n T ~ H , K ( y i ⋅ H ′ ) = T ~ H , K ( ∏ i = 1 n y i ⋅ H ′ ) = T ~ H , K ( ∏ i = 1 n u x ( i ) ⋅ H ′ ) = T ~ H , K ( T G , H ( x ) ) {\displaystyle {\begin{aligned}T_{G,K}(x)&=\prod _{i=1}^{n}\prod _{j=1}^{m}v_{y_{i}}(j)\cdot K'\\\&=\prod _{i=1}^{n}\prod _{j=1}^{m}h_{\sigma _{y_{i}}(j)}^{-1}g_{\pi _{x}(i)}^{-1}xg_{i}h_{j}\cdot K'\\\&=\prod _{i=1}^{n}\prod _{j=1}^{m}h_{\sigma _{y_{i}}(j)}^{-1}u_{x}(i)h_{j}\cdot K'\\\&=\prod _{i=1}^{n}\prod _{j=1}^{m}h_{\sigma _{y_{i}}(j)}^{-1}y_{i}h_{j}\cdot K'\\\&=\prod _{i=1}^{n}{\tilde {T}}_{H,K}\left(y_{i}\cdot H'\right)\\\&={\tilde {T}}_{H,K}\left(\prod _{i=1}^{n}y_{i}\cdot H'\right)\\\&={\tilde {T}}_{H,K}\left(\prod _{i=1}^{n}u_{x}(i)\cdot H'\right)\\\&={\tilde {T}}_{H,K}(T_{G,H}(x))\end{aligned}}} Finally, we want to emphasize the structural peculiarity of the monomial representation { G -> K n ⋅ m × S n ⋅ m x ↦ ( k x ( 1 , 1 ) , … , k x ( n , m ) ; γ x ) {\displaystyle {\begin{cases}G\to K^{n\cdot m}\times S_{n\cdot m}\\\x\mapsto (k_{x}(1,1),\ldots ,k_{x}(n,m);\gamma _{x})\end{cases}}} which corresponds to the composite of Artin transfers, defining k x ( i , j ) := ( ( g h ) γ x ( i , j ) ) − 1 x ( g h ) ( i , j ) ∈ K {\displaystyle k_{x}(i,j):=\left((gh)_{\gamma _{x}(i,j)}\right)^{-1}x(gh)_{(i,j)}\in K} for a permutation γ x ∈ S n ⋅ m {\displaystyle \gamma _{x}\in S_{n\cdot m}} , and using the symbolic notation ( g h ) ( i , j ) := g i h j {\displaystyle (gh)_{(i,j)}:=g_{i}h_{j}} for all pairs of subscripts 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} , 1 ≤ j ≤ m {\displaystyle 1\leq j\leq m} . The preceding proof has shown that k x ( i , j ) = h σ y i ( j ) − 1 g π x ( i ) − 1 x g i h j . {\displaystyle k_{x}(i,j)=h_{\sigma _{y_{i}}(j)}^{-1}g_{\pi _{x}(i)}^{-1}xg_{i}h_{j}.} Therefore, the action of the permutation γ x {\displaystyle \gamma _{x}} on the set [ 1 , n ] × [ 1 , m ] {\displaystyle [1,n]\times [1,m]} is given by γ x ( i , j ) = ( π x ( i ) , σ u x ( i ) ( j ) ) {\displaystyle \gamma _{x}(i,j)=(\pi _{x}(i),\sigma _{u_{x}(i)}(j))} . The action on the second component j {\displaystyle j} depends on the first component i {\displaystyle i} (via the permutation σ u x ( i ) ∈ S m {\displaystyle \sigma _{u_{x}(i)}\in S_{m}} ), whereas the action on the first component i {\displaystyle i} is independent of the second component j {\displaystyle j} . Therefore, the permutation γ x ∈ S n ⋅ m {\displaystyle \gamma _{x}\in S_{n\cdot m}} can be identified with the multiplet ( π x ; σ u x ( 1 ) , … , σ u x ( n ) ) ∈ S n × S m n , {\displaystyle (\pi _{x};\sigma _{u_{x}(1)},\ldots ,\sigma _{u_{x}(n)})\in S_{n}\times S_{m}^{n},} which will be written in twisted form in the next section. ### Wreath product of S(m) and S(n)[edit] The permutations γ x {\displaystyle \gamma _{x}} , which arose as second components of the monomial representation { G -> K ≀ S n ⋅ m x ↦ ( k x ( 1 , 1 ) , … , k x ( n , m ) ; γ x ) {\displaystyle {\begin{cases}G\to K\wr S_{n\cdot m}\\\x\mapsto (k_{x}(1,1),\ldots ,k_{x}(n,m);\gamma _{x})\end{cases}}} in the previous section, are of a very special kind. They belong to the stabilizer of the natural equipartition of the set [ 1 , n ] × [ 1 , m ] {\displaystyle [1,n]\times [1,m]} into the n {\displaystyle n} rows of the corresponding matrix (rectangular array). Using the peculiarities of the composition of Artin transfers in the previous section, we show that this stabilizer is isomorphic to the wreath product S m ≀ S n {\displaystyle S_{m}\wr S_{n}} of the symmetric groups S m {\displaystyle S_{m}} and S n {\displaystyle S_{n}} with respect to the set { 1 , … , n } {\displaystyle \\{1,\ldots ,n\\}} , whose underlying set S m n × S n {\displaystyle S_{m}^{n}\times S_{n}} is endowed with the following law of composition: ( 10 ) ∀ x , z ∈ G : γ x ⋅ γ z = ( σ u x ( 1 ) , … , σ u x ( n ) ; π x ) ⋅ ( σ u z ( 1 ) , … , σ u z ( n ) ; π z ) = ( σ u x ( π z ( 1 ) ) ∘ σ u z ( 1 ) , … , σ u x ( π z ( n ) ) ∘ σ u z ( n ) ; π x ∘ π z ) = ( σ u x z ( 1 ) , … , σ u x z ( n ) ; π x z ) = γ x z {\displaystyle {\begin{aligned}(10)\quad \forall x,z\in G:\qquad \gamma _{x}\cdot \gamma _{z}&=(\sigma _{u_{x}(1)},\ldots ,\sigma _{u_{x}(n)};\pi _{x})\cdot (\sigma _{u_{z}(1)},\ldots ,\sigma _{u_{z}(n)};\pi _{z})\\\&=(\sigma _{u_{x}(\pi _{z}(1))}\circ \sigma _{u_{z}(1)},\ldots ,\sigma _{u_{x}(\pi _{z}(n))}\circ \sigma _{u_{z}(n)};\pi _{x}\circ \pi _{z})\\\&=(\sigma _{u_{xz}(1)},\ldots ,\sigma _{u_{xz}(n)};\pi _{xz})\\\&=\gamma _{xz}\end{aligned}}} This law reminds of the chain rule D ( g ∘ f ) ( x ) = D ( g ) ( f ( x ) ) ∘ D ( f ) ( x ) {\displaystyle D(g\circ f)(x)=D(g)(f(x))\circ D(f)(x)} for the Fréchet derivative in x ∈ E {\displaystyle x\in E} of the compositum of differentiable functions f : E -> F {\displaystyle f:E\to F} and g : F -> G {\displaystyle g:F\to G} between complete normed spaces. The above considerations establish a third representation, the stabilizer representation, { G -> S m ≀ S n x ↦ ( σ u x ( 1 ) , … , σ u x ( n ) ; π x ) {\displaystyle {\begin{cases}G\to S_{m}\wr S_{n}\\\x\mapsto (\sigma _{u_{x}(1)},\ldots ,\sigma _{u_{x}(n)};\pi _{x})\end{cases}}} of the group G {\displaystyle G} in the wreath product S m ≀ S n {\displaystyle S_{m}\wr S_{n}} , similar to the permutation representation and the monomial representation. As opposed to the latter, the stabilizer representation cannot be injective, in general. For instance, certainly not, if G {\displaystyle G} is infinite. Formula (10) proves the following statement. Theorem. The stabilizer representation { G -> S m ≀ S n x ↦ γ x = ( σ u x ( 1 ) , … , σ u x ( n ) ; π x ) {\displaystyle {\begin{cases}G\to S_{m}\wr S_{n}\\\x\mapsto \gamma _{x}=(\sigma _{u_{x}(1)},\ldots ,\sigma _{u_{x}(n)};\pi _{x})\end{cases}}} of the group G {\displaystyle G} in the wreath product S m ≀ S n {\displaystyle S_{m}\wr S_{n}} of symmetric groups is a group homomorphism. ### Cycle decomposition[edit] Let ( g 1 , … , g n ) {\displaystyle (g_{1},\ldots ,g_{n})} be a left transversal of a subgroup H {\displaystyle H} of finite index n {\displaystyle n} in a group G {\displaystyle G} and x ↦ π x {\displaystyle x\mapsto \pi _{x}} be its associated permutation representation. Theorem.[1][3][4][5][8][9] Suppose the permutation π x {\displaystyle \pi _{x}} decomposes into pairwise disjoint (and thus commuting) cycles ζ 1 , … , ζ t ∈ S n {\displaystyle \zeta _{1},\ldots ,\zeta _{t}\in S_{n}} of lengths f 1 , … f t , {\displaystyle f_{1},\ldots f_{t},} which is unique up to the ordering of the cycles. More explicitly, suppose ( 11 ) ( g j H , g ζ j ( j ) H , g ζ j 2 ( j ) H , … , g ζ j f j − 1 ( j ) H ) = ( g j H , x g j H , x 2 g j H , … , x f j − 1 g j H ) , {\displaystyle (11)\quad \left(g_{j}H,g_{\zeta _{j}(j)}H,g_{\zeta _{j}^{2}(j)}H,\ldots ,g_{\zeta _{j}^{f_{j}-1}(j)}H\right)=\left(g_{j}H,xg_{j}H,x^{2}g_{j}H,\ldots ,x^{f_{j}-1}g_{j}H\right),} for 1 ≤ j ≤ t {\displaystyle 1\leq j\leq t} , and f 1 + ⋯ + f t = n . {\displaystyle f_{1}+\cdots +f_{t}=n.} Then the image of x ∈ G {\displaystyle x\in G} under the Artin transfer is given by ( 12 ) T G , H ( x ) = ∏ j = 1 t g j − 1 x f j g j ⋅ H ′ . {\displaystyle (12)\quad T_{G,H}(x)=\prod _{j=1}^{t}g_{j}^{-1}x^{f_{j}}g_{j}\cdot H'.} Proof Define ℓ j , k := x k g j {\displaystyle \ell _{j,k}:=x^{k}g_{j}} for 0 ≤ k ≤ f j − 1 {\displaystyle 0\leq k\leq f_{j}-1} and 1 ≤ j ≤ t {\displaystyle 1\leq j\leq t} . This is a left transversal of H {\displaystyle H} in G {\displaystyle G} since ( 13 ) G = ⨆ j = 1 t ⨆ k = 0 f j − 1 x k g j H {\displaystyle (13)\quad G=\bigsqcup _{j=1}^{t}\bigsqcup _{k=0}^{f_{j}-1}x^{k}g_{j}H} is a disjoint decomposition of G {\displaystyle G} into left cosets of H {\displaystyle H} . Fix a value of 1 ≤ j ≤ t {\displaystyle 1\leq j\leq t} . Then: x ℓ j , k = x x k g j = x k + 1 g j = ℓ j , k + 1 ∈ ℓ j , k + 1 H ∀ k ∈ { 0 , … , f j − 2 } x ℓ j , f j − 1 = x x f j − 1 g j = x f j g j ∈ g j H = ℓ j , 0 H {\displaystyle {\begin{aligned}x\ell _{j,k}&=xx^{k}g_{j}=x^{k+1}g_{j}=\ell _{j,k+1}\in \ell _{j,k+1}H&&\forall k\in \\{0,\ldots ,f_{j}-2\\}\\\x\ell _{j,f_{j}-1}&=xx^{f_{j}-1}g_{j}=x^{f_{j}}g_{j}\in g_{j}H=\ell _{j,0}H\end{aligned}}} Define: u x ( j , k ) := ℓ j , k + 1 − 1 x ℓ j , k = 1 ∈ H ∀ k ∈ { 0 , … , f j − 2 } u x ( j , f j − 1 ) := ℓ j , 0 − 1 x ℓ j , f j − 1 = g j − 1 x f j g j ∈ H {\displaystyle {\begin{aligned}u_{x}(j,k)&:=\ell _{j,k+1}^{-1}x\ell _{j,k}=1\in H&&\forall k\in \\{0,\ldots ,f_{j}-2\\}\\\u_{x}(j,f_{j}-1)&:=\ell _{j,0}^{-1}x\ell _{j,f_{j}-1}=g_{j}^{-1}x^{f_{j}}g_{j}\in H\end{aligned}}} Consequently, T G , H ( x ) = ∏ j = 1 t ∏ k = 0 f j − 1 u x ( j , k ) ⋅ H ′ = ∏ j = 1 t ( ∏ k = 0 f j − 2 1 ) ⋅ u x ( j , f j − 1 ) ⋅ H ′ = ∏ j = 1 t g j − 1 x f j g j ⋅ H ′ . {\displaystyle T_{G,H}(x)=\prod _{j=1}^{t}\prod _{k=0}^{f_{j}-1}u_{x}(j,k)\cdot H'=\prod _{j=1}^{t}\left(\prod _{k=0}^{f_{j}-2}1\right)\cdot u_{x}(j,f_{j}-1)\cdot H'=\prod _{j=1}^{t}g_{j}^{-1}x^{f_{j}}g_{j}\cdot H'.} The cycle decomposition corresponds to a ( ⟨ x ⟩ , H ) {\displaystyle (\langle x\rangle ,H)} double coset decomposition of G {\displaystyle G} : G = ⨆ j = 1 t ⟨ x ⟩ g j H {\displaystyle G=\bigsqcup _{j=1}^{t}\langle x\rangle g_{j}H} It was this cycle decomposition form of the transfer homomorphism which was given by E. Artin in his original 1929 paper.[3] ### Transfer to a normal subgroup[edit] Let H {\displaystyle H} be a normal subgroup of finite index n {\displaystyle n} in a group G {\displaystyle G} . Then we have x H = H x {\displaystyle xH=Hx} , for all x ∈ G {\displaystyle x\in G} , and there exists the quotient group G / H {\displaystyle G/H} of order n {\displaystyle n} . For an element x ∈ G {\displaystyle x\in G} , we let f := o r d ( x H ) {\displaystyle f:=\mathrm {ord} (xH)} denote the order of the coset x H {\displaystyle xH} in G / H {\displaystyle G/H} , and we let ( g 1 , … , g t ) {\displaystyle (g_{1},\ldots ,g_{t})} be a left transversal of the subgroup ⟨ x , H ⟩ {\displaystyle \langle x,H\rangle } in G {\displaystyle G} , where t = n / f {\displaystyle t=n/f} . Theorem. Then the image of x ∈ G {\displaystyle x\in G} under the Artin transfer T G , H {\displaystyle T_{G,H}} is given by: ( 14 ) T G , H ( x ) = ∏ j = 1 t g j − 1 x f g j ⋅ H ′ {\displaystyle (14)\quad T_{G,H}(x)=\prod _{j=1}^{t}g_{j}^{-1}x^{f}g_{j}\cdot H'} . Proof ⟨ x H ⟩ {\displaystyle \langle xH\rangle } is a cyclic subgroup of order f {\displaystyle f} in G / H {\displaystyle G/H} , and a left transversal ( g 1 , … , g t ) {\displaystyle (g_{1},\ldots ,g_{t})} of the subgroup ⟨ x , H ⟩ {\displaystyle \langle x,H\rangle } in G {\displaystyle G} , where t = n / f {\displaystyle t=n/f} and G = ⊔ j = 1 t g j ⟨ x , H ⟩ {\displaystyle G=\sqcup _{j=1}^{t}g_{j}\langle x,H\rangle } is the corresponding disjoint left coset decomposition, can be refined to a left transversal g j x k ( 1 ≤ j ≤ t , 0 ≤ k ≤ f − 1 ) {\displaystyle g_{j}x^{k}(1\leq j\leq t,\ 0\leq k\leq f-1)} with disjoint left coset decomposition: ( 15 ) G = ⊔ j = 1 t ⊔ k = 0 f − 1 g j x k H {\displaystyle (15)\quad G=\sqcup _{j=1}^{t}\sqcup _{k=0}^{f-1}g_{j}x^{k}H} of H {\displaystyle H} in G {\displaystyle G} . Hence, the formula for the image of x {\displaystyle x} under the Artin transfer T G , H {\displaystyle T_{G,H}} in the previous section takes the particular shape T G , H ( x ) = ∏ j = 1 t g j − 1 x f g j ⋅ H ′ {\displaystyle T_{G,H}(x)=\prod _{j=1}^{t}g_{j}^{-1}x^{f}g_{j}\cdot H'} with exponent f {\displaystyle f} independent of j {\displaystyle j} . Corollary. In particular, the inner transfer of an element x ∈ H {\displaystyle x\in H} is given as a symbolic power: ( 16 ) T G , H ( x ) = x T r G ( H ) ⋅ H ′ {\displaystyle (16)\quad T_{G,H}(x)=x^{\mathrm {Tr} _{G}(H)}\cdot H'} with the trace element ( 17 ) T r G ( H ) = ∑ j = 1 t g j ∈ Z [ G ] {\displaystyle (17)\quad \mathrm {Tr} _{G}(H)=\sum _{j=1}^{t}g_{j}\in \mathbb {Z} [G]} of H {\displaystyle H} in G {\displaystyle G} as symbolic exponent. The other extreme is the outer transfer of an element x ∈ G ∖ H {\displaystyle x\in G\setminus H} which generates G / H {\displaystyle G/H} , that is G = ⟨ x , H ⟩ {\displaystyle G=\langle x,H\rangle } . It is simply an n {\displaystyle n} th power ( 18 ) T G , H ( x ) = x n ⋅ H ′ {\displaystyle (18)\quad T_{G,H}(x)=x^{n}\cdot H'} . Proof The inner transfer of an element x ∈ H {\displaystyle x\in H} , whose coset x H = H {\displaystyle xH=H} is the principal set in G / H {\displaystyle G/H} of order f = 1 {\displaystyle f=1} , is given as the symbolic power T G , H ( x ) = ∏ j = 1 t g j − 1 x g j ⋅ H ′ = ∏ j = 1 t x g j ⋅ H ′ = x ∑ j = 1 t g j ⋅ H ′ {\displaystyle T_{G,H}(x)=\prod _{j=1}^{t}g_{j}^{-1}xg_{j}\cdot H'=\prod _{j=1}^{t}x^{g_{j}}\cdot H'=x^{\sum _{j=1}^{t}g_{j}}\cdot H'} with the trace element T r G ( H ) = ∑ j = 1 t g j ∈ Z [ G ] {\displaystyle \mathrm {Tr} _{G}(H)=\sum _{j=1}^{t}g_{j}\in \mathbb {Z} [G]} of H {\displaystyle H} in G {\displaystyle G} as symbolic exponent. The outer transfer of an element x ∈ G ∖ H {\displaystyle x\in G\setminus H} which generates G / H {\displaystyle G/H} , that is G = ⟨ x , H ⟩ {\displaystyle G=\langle x,H\rangle } , whence the coset x H {\displaystyle xH} is generator of G / H {\displaystyle G/H} with order f = n {\displaystyle f=n} , is given as the n {\displaystyle n} th power T G , H ( x ) = ∏ j = 1 1 1 − 1 ⋅ x n ⋅ 1 ⋅ H ′ = x n ⋅ H ′ . {\displaystyle T_{G,H}(x)=\prod _{j=1}^{1}1^{-1}\cdot x^{n}\cdot 1\cdot H'=x^{n}\cdot H'.} Transfers to normal subgroups will be the most important cases in the sequel, since the central concept of this article, the Artin pattern, which endows descendant trees with additional structure, consists of targets and kernels of Artin transfers from a group G {\displaystyle G} to intermediate groups G ′ ≤ H ≤ G {\displaystyle G'\leq H\leq G} between G {\displaystyle G} and G ′ {\displaystyle G'} . For these intermediate groups we have the following lemma. Lemma. All subgroups containing the commutator subgroup are normal. Proof Let G ′ ≤ H ≤ G {\displaystyle G'\leq H\leq G} . If H {\displaystyle H} were not a normal subgroup of G {\displaystyle G} , then we had x − 1 H x ⊈ H {\displaystyle x^{-1}Hx\not \subseteq H} for some element x ∈ G ∖ H {\displaystyle x\in G\setminus H} . This would imply the existence of elements h ∈ H {\displaystyle h\in H} and y ∈ G ∖ H {\displaystyle y\in G\setminus H} such that x − 1 h x = y {\displaystyle x^{-1}hx=y} , and consequently the commutator [ h , x ] = h − 1 x − 1 h x = h − 1 y {\displaystyle [h,x]=h^{-1}x^{-1}hx=h^{-1}y} would be an element in G ∖ H {\displaystyle G\setminus H} in contradiction to G ′ ≤ H {\displaystyle G'\leq H} . Explicit implementations of Artin transfers in the simplest situations are presented in the following section. ## Computational implementation[edit] ### Abelianization of type (p,p)[edit] Let G {\displaystyle G} be a p-group with abelianization G / G ′ {\displaystyle G/G'} of elementary abelian type ( p , p ) {\displaystyle (p,p)} . Then G {\displaystyle G} has p + 1 {\displaystyle p+1} maximal subgroups H 1 , … , H p + 1 {\displaystyle H_{1},\ldots ,H_{p+1}} of index p . {\displaystyle p.} Lemma. In this particular case, the Frattini subgroup, which is defined as the intersection of all maximal subgroups coincides with the commutator subgroup. Proof. To see this note that due to the abelian type of G / G ′ {\displaystyle G/G'} the commutator subgroup contains all p-th powers G ′ ⊃ G p , {\displaystyle G'\supset G^{p},} and thus we have Φ ( G ) = G p ⋅ G ′ = G ′ {\displaystyle \Phi (G)=G^{p}\cdot G'=G'} . For each 1 ≤ i ≤ p + 1 {\displaystyle 1\leq i\leq p+1} , let T i : G -> H i / H i ′ {\displaystyle T_{i}:G\to H_{i}/H_{i}'} be the Artin transfer homomorphism. According to Burnside's basis theorem the group G {\displaystyle G} can therefore be generated by two elements x , y {\displaystyle x,y} such that x p , y p ∈ G ′ . {\displaystyle x^{p},y^{p}\in G'.} For each of the maximal subgroups H i {\displaystyle H_{i}} , which are also normal we need a generator h i {\displaystyle h_{i}} with respect to G ′ {\displaystyle G'} , and a generator t i {\displaystyle t_{i}} of a transversal ( 1 , t i , t i 2 , … , t i p − 1 ) {\displaystyle (1,t_{i},t_{i}^{2},\ldots ,t_{i}^{p-1})} such that H i = ⟨ h i , G ′ ⟩ G = ⟨ t i , H i ⟩ = ⨆ j = 0 p − 1 t i j H i {\displaystyle {\begin{aligned}H_{i}&=\langle h_{i},G'\rangle \\\G&=\langle t_{i},H_{i}\rangle =\bigsqcup _{j=0}^{p-1}t_{i}^{j}H_{i}\end{aligned}}} A convenient selection is given by ( 19 ) { h 1 = y t 1 = x h i = x y i − 2 2 ≤ i ≤ p + 1 t i = y 2 ≤ i ≤ p + 1 {\displaystyle (19)\quad {\begin{cases}h_{1}=y\\\t_{1}=x\\\h_{i}=xy^{i-2}&2\leq i\leq p+1\\\t_{i}=y&2\leq i\leq p+1\end{cases}}} Then, for each 1 ≤ i ≤ p + 1 {\displaystyle 1\leq i\leq p+1} we use equations (16) and (18) to implement the inner and outer transfers: ( 20 ) T i ( h i ) = h i T r G ( H i ) ⋅ H i ′ = h i 1 + t i + t i 2 + ⋯ + t i p − 1 ⋅ H i ′ = h i ⋅ ( t i − 1 h i t i ) ⋅ ( t i − 2 h i t i 2 ) ⋯ ( t i − p + 1 h i t i p − 1 ) ⋅ H i ′ = ( h i t i − 1 ) p t i p ⋅ H i ′ ( 21 ) T i ( t i ) = t i p ⋅ H i ′ {\displaystyle {\begin{aligned}(20)\quad T_{i}(h_{i})&=h_{i}^{\mathrm {Tr} _{G}(H_{i})}\cdot H_{i}'=h_{i}^{1+t_{i}+t_{i}^{2}+\cdots +t_{i}^{p-1}}\cdot H_{i}'=h_{i}\cdot \left(t_{i}^{-1}h_{i}t_{i}\right)\cdot \left(t_{i}^{-2}h_{i}t_{i}^{2}\right)\cdots \left(t_{i}^{-p+1}h_{i}t_{i}^{p-1}\right)\cdot H_{i}'=\left(h_{i}t_{i}^{-1}\right)^{p}t_{i}^{p}\cdot H_{i}'\\\\(21)\quad T_{i}(t_{i})&=t_{i}^{p}\cdot H_{i}'\end{aligned}}} , The reason is that in G / H i , {\displaystyle G/H_{i},} o r d ( h i H i ) = 1 {\displaystyle \mathrm {ord} (h_{i}H_{i})=1} and o r d ( t i H i ) = p . {\displaystyle \mathrm {ord} (t_{i}H_{i})=p.} The complete specification of the Artin transfers T i {\displaystyle T_{i}} also requires explicit knowledge of the derived subgroups H i ′ {\displaystyle H_{i}'} . Since G ′ {\displaystyle G'} is a normal subgroup of index p {\displaystyle p} in H i {\displaystyle H_{i}} , a certain general reduction is possible by H i ′ = [ H i , H i ] = [ G ′ , H i ] = ( G ′ ) h i − 1 , {\displaystyle H_{i}'=[H_{i},H_{i}]=[G',H_{i}]=(G')^{h_{i}-1},} [10] but a presentation of G {\displaystyle G} must be known for determining generators of G ′ = ⟨ s 1 , … , s n ⟩ {\displaystyle G'=\langle s_{1},\ldots ,s_{n}\rangle } , whence ( 22 ) H i ′ = ( G ′ ) h i − 1 = ⟨ [ s 1 , h i ] , … , [ s n , h i ] ⟩ . {\displaystyle (22)\quad H_{i}'=(G')^{h_{i}-1}=\langle [s_{1},h_{i}],\ldots ,[s_{n},h_{i}]\rangle .} ### Abelianization of type (p2,p)[edit] Let G {\displaystyle G} be a p-group with abelianization G / G ′ {\displaystyle G/G'} of non-elementary abelian type ( p 2 , p ) {\displaystyle (p^{2},p)} . Then G {\displaystyle G} has p + 1 {\displaystyle p+1} maximal subgroups H 1 , … , H p + 1 {\displaystyle H_{1},\ldots ,H_{p+1}} of index p {\displaystyle p} and p + 1 {\displaystyle p+1} subgroups U 1 , … , U p + 1 {\displaystyle U_{1},\ldots ,U_{p+1}} of index p 2 . {\displaystyle p^{2}.} For each i ∈ { 1 , … , p + 1 } {\displaystyle i\in \\{1,\ldots ,p+1\\}} let T 1 , i : G -> H i / H i ′ T 2 , i : G -> U i / U i ′ {\displaystyle {\begin{aligned}T_{1,i}:G&\to H_{i}/H_{i}'\\\T_{2,i}:G&\to U_{i}/U_{i}'\end{aligned}}} be the Artin transfer homomorphisms. Burnside's basis theorem asserts that the group G {\displaystyle G} can be generated by two elements x , y {\displaystyle x,y} such that x p 2 , y p ∈ G ′ . {\displaystyle x^{p^{2}},y^{p}\in G'.} We begin by considering the first layer of subgroups. For each of the normal subgroups H i {\displaystyle H_{i}} , we select a generator ( 23 ) h i = x y i − 1 {\displaystyle (23)\quad h_{i}=xy^{i-1}} such that H i = ⟨ h i , G ′ ⟩ {\displaystyle H_{i}=\langle h_{i},G'\rangle } . These are the cases where the factor group H i / G ′ {\displaystyle H_{i}/G'} is cyclic of order p 2 {\displaystyle p^{2}} . However, for the distinguished maximal subgroup H p + 1 {\displaystyle H_{p+1}} , for which the factor group H p + 1 / G ′ {\displaystyle H_{p+1}/G'} is bicyclic of type ( p , p ) {\displaystyle (p,p)} , we need two generators: ( 24 ) { h p + 1 = y h 0 = x p {\displaystyle (24)\quad {\begin{cases}h_{p+1}=y\\\h_{0}=x^{p}\end{cases}}} such that H p + 1 = ⟨ h p + 1 , h 0 , G ′ ⟩ {\displaystyle H_{p+1}=\langle h_{p+1},h_{0},G'\rangle } . Further, a generator t i {\displaystyle t_{i}} of a transversal must be given such that G = ⟨ t i , H i ⟩ {\displaystyle G=\langle t_{i},H_{i}\rangle } , for each 1 ≤ i ≤ p + 1 {\displaystyle 1\leq i\leq p+1} . It is convenient to define ( 25 ) { t i = y 1 ≤ i ≤ p t p + 1 = x {\displaystyle (25)\quad {\begin{cases}t_{i}=y&1\leq i\leq p\\\t_{p+1}=x\end{cases}}} Then, for each 1 ≤ i ≤ p + 1 {\displaystyle 1\leq i\leq p+1} , we have inner and outer transfers: ( 26 ) T 1 , i ( h i ) = h i T r G ( H i ) ⋅ H i ′ = h i 1 + t i + t i 2 + … + t i p − 1 ⋅ H i ′ = ( h i t i − 1 ) p t i p ⋅ H i ′ ( 27 ) T 1 , i ( t i ) = t i p ⋅ H i ′ {\displaystyle {\begin{aligned}(26)\quad T_{1,i}(h_{i})&=h_{i}^{\mathrm {Tr} _{G}(H_{i})}\cdot H_{i}'=h_{i}^{1+t_{i}+t_{i}^{2}+\ldots +t_{i}^{p-1}}\cdot H_{i}'=\left(h_{i}t_{i}^{-1}\right)^{p}t_{i}^{p}\cdot H_{i}'\\\\(27)\quad T_{1,i}(t_{i})&=t_{i}^{p}\cdot H_{i}'\end{aligned}}} since o r d ( h i H i ) = 1 {\displaystyle \mathrm {ord} (h_{i}H_{i})=1} and o r d ( t i H i ) = p {\displaystyle \mathrm {ord} (t_{i}H_{i})=p} . Now we continue by considering the second layer of subgroups. For each of the normal subgroups U i {\displaystyle U_{i}} , we select a generator ( 28 ) { u 1 = y u i = x p y i − 1 2 ≤ i ≤ p u p + 1 = x p {\displaystyle (28)\quad {\begin{cases}u_{1}=y\\\u_{i}=x^{p}y^{i-1}&2\leq i\leq p\\\u_{p+1}=x^{p}\end{cases}}} such that U i = ⟨ u i , G ′ ⟩ {\displaystyle U_{i}=\langle u_{i},G'\rangle } . Among these subgroups, the Frattini subgroup U p + 1 = ⟨ x p , G ′ ⟩ = G p ⋅ G ′ {\displaystyle U_{p+1}=\langle x^{p},G'\rangle =G^{p}\cdot G'} is particularly distinguished. A uniform way of defining generators t i , w i {\displaystyle t_{i},w_{i}} of a transversal such that G = ⟨ t i , w i , U i ⟩ {\displaystyle G=\langle t_{i},w_{i},U_{i}\rangle } , is to set ( 29 ) { t i = x 1 ≤ i ≤ p w i = x p 1 ≤ i ≤ p t p + 1 = x w p + 1 = y {\displaystyle (29)\quad {\begin{cases}t_{i}=x&1\leq i\leq p\\\w_{i}=x^{p}&1\leq i\leq p\\\t_{p+1}=x\\\w_{p+1}=y\end{cases}}} Since o r d ( u i U i ) = 1 {\displaystyle \mathrm {ord} (u_{i}U_{i})=1} , but on the other hand o r d ( t i U i ) = p 2 {\displaystyle \mathrm {ord} (t_{i}U_{i})=p^{2}} and o r d ( w i U i ) = p {\displaystyle \mathrm {ord} (w_{i}U_{i})=p} , for 1 ≤ i ≤ p + 1 {\displaystyle 1\leq i\leq p+1} , with the single exception that o r d ( t p + 1 U p + 1 ) = p {\displaystyle \mathrm {ord} (t_{p+1}U_{p+1})=p} , we obtain the following expressions for the inner and outer transfers ( 30 ) T 2 , i ( u i ) = u i T r G ( U i ) ⋅ U i ′ = u i ∑ j = 0 p − 1 ∑ k = 0 p − 1 w i j t i k ⋅ U i ′ = ∏ j = 0 p − 1 ∏ k = 0 p − 1 ( w i j t i k ) − 1 u i w i j t i k ⋅ U i ′ ( 31 ) T 2 , i ( t i ) = t i p 2 ⋅ U i ′ {\displaystyle {\begin{aligned}(30)\quad T_{2,i}(u_{i})&=u_{i}^{\mathrm {Tr} _{G}(U_{i})}\cdot U_{i}'=u_{i}^{\sum _{j=0}^{p-1}\sum _{k=0}^{p-1}w_{i}^{j}t_{i}^{k}}\cdot U_{i}'=\prod _{j=0}^{p-1}\prod _{k=0}^{p-1}(w_{i}^{j}t_{i}^{k})^{-1}u_{i}w_{i}^{j}t_{i}^{k}\cdot U_{i}'\\\\(31)\quad T_{2,i}(t_{i})&=t_{i}^{p^{2}}\cdot U_{i}'\end{aligned}}} exceptionally ( 32 ) T 2 , p + 1 ( t p + 1 ) = ( t p + 1 p ) 1 + w p + 1 + w p + 1 2 + … + w p + 1 p − 1 ⋅ U p + 1 ′ ( 33 ) T 2 , i ( w i ) = ( w i p ) 1 + t i + t i 2 + … + t i p − 1 ⋅ U i ′ 1 ≤ i ≤ p + 1 {\displaystyle {\begin{aligned}&(32)\quad T_{2,p+1}\left(t_{p+1}\right)=\left(t_{p+1}^{p}\right)^{1+w_{p+1}+w_{p+1}^{2}+\ldots +w_{p+1}^{p-1}}\cdot U_{p+1}'\\\&(33)\quad T_{2,i}(w_{i})=\left(w_{i}^{p}\right)^{1+t_{i}+t_{i}^{2}+\ldots +t_{i}^{p-1}}\cdot U_{i}'&&1\leq i\leq p+1\end{aligned}}} The structure of the derived subgroups H i ′ {\displaystyle H_{i}'} and U i ′ {\displaystyle U_{i}'} must be known to specify the action of the Artin transfers completely. ## Transfer kernels and targets[edit] Let G {\displaystyle G} be a group with finite abelianization G / G ′ {\displaystyle G/G'} . Suppose that ( H i ) i ∈ I {\displaystyle (H_{i})_{i\in I}} denotes the family of all subgroups which contain G ′ {\displaystyle G'} and are therefore necessarily normal, enumerated by a finite index set I {\displaystyle I} . For each i ∈ I {\displaystyle i\in I} , let T i := T G , H i {\displaystyle T_{i}:=T_{G,H_{i}}} be the Artin transfer from G {\displaystyle G} to the abelianization H i / H i ′ {\displaystyle H_{i}/H_{i}'} . Definition.[11] The family of normal subgroups ϰ H ( G ) = ( ker ⁡ ( T i ) ) i ∈ I {\displaystyle \varkappa _{H}(G)=(\ker(T_{i}))_{i\in I}} is called the transfer kernel type (TKT) of G {\displaystyle G} with respect to ( H i ) i ∈ I {\displaystyle (H_{i})_{i\in I}} , and the family of abelianizations (resp. their abelian type invariants) τ H ( G ) = ( H i / H i ′ ) i ∈ I {\displaystyle \tau _{H}(G)=(H_{i}/H_{i}')_{i\in I}} is called the transfer target type (TTT) of G {\displaystyle G} with respect to ( H i ) i ∈ I {\displaystyle (H_{i})_{i\in I}} . Both families are also called multiplets whereas a single component will be referred to as a singulet. Important examples for these concepts are provided in the following two sections. ## Abelianization of type (p,p)[edit] Let G {\displaystyle G} be a p-group with abelianization G / G ′ {\displaystyle G/G'} of elementary abelian type ( p , p ) {\displaystyle (p,p)} . Then G {\displaystyle G} has p + 1 {\displaystyle p+1} maximal subgroups H 1 , … , H p + 1 {\displaystyle H_{1},\ldots ,H_{p+1}} of index p {\displaystyle p} . For i ∈ { 1 , … , p + 1 } {\displaystyle i\in \\{1,\ldots ,p+1\\}} let T i : G -> H i / H i ′ {\displaystyle T_{i}:G\to H_{i}/H_{i}'} denote the Artin transfer homomorphism. Definition. The family of normal subgroups ϰ H ( G ) = ( ker ⁡ ( T i ) ) 1 ≤ i ≤ p + 1 {\displaystyle \varkappa _{H}(G)=(\ker(T_{i}))_{1\leq i\leq p+1}} is called the transfer kernel type (TKT) of G {\displaystyle G} with respect to H 1 , … , H p + 1 {\displaystyle H_{1},\ldots ,H_{p+1}} . Remark. For brevity, the TKT is identified with the multiplet ( ϰ ( i ) ) 1 ≤ i ≤ p + 1 {\displaystyle (\varkappa (i))_{1\leq i\leq p+1}} , whose integer components are given by ϰ ( i ) = { 0 ker ⁡ ( T i ) = G j ker ⁡ ( T i ) = H j for some 1 ≤ j ≤ p + 1 {\displaystyle \varkappa (i)={\begin{cases}0&\ker(T_{i})=G\\\j&\ker(T_{i})=H_{j}{\text{ for some }}1\leq j\leq p+1\end{cases}}} Here, we take into consideration that each transfer kernel ker ⁡ ( T i ) {\displaystyle \ker(T_{i})} must contain the commutator subgroup G ′ {\displaystyle G'} of G {\displaystyle G} , since the transfer target H i / H i ′ {\displaystyle H_{i}/H_{i}'} is abelian. However, the minimal case ker ⁡ ( T i ) = G ′ {\displaystyle \ker(T_{i})=G'} cannot occur. Remark. A renumeration of the maximal subgroups K i = H π ( i ) {\displaystyle K_{i}=H_{\pi (i)}} and of the transfers V i = T π ( i ) {\displaystyle V_{i}=T_{\pi (i)}} by means of a permutation π ∈ S p + 1 {\displaystyle \pi \in S_{p+1}} gives rise to a new TKT λ K ( G ) = ( ker ⁡ ( V i ) ) 1 ≤ i ≤ p + 1 {\displaystyle \lambda _{K}(G)=(\ker(V_{i}))_{1\leq i\leq p+1}} with respect to K 1 , … , K p + 1 {\displaystyle K_{1},\ldots ,K_{p+1}} , identified with ( λ ( i ) ) 1 ≤ i ≤ p + 1 {\displaystyle (\lambda (i))_{1\leq i\leq p+1}} , where λ ( i ) = { 0 ker ⁡ ( V i ) = G j ker ⁡ ( V i ) = K j for some 1 ≤ j ≤ p + 1 {\displaystyle \lambda (i)={\begin{cases}0&\ker(V_{i})=G\\\j&\ker(V_{i})=K_{j}{\text{ for some }}1\leq j\leq p+1\end{cases}}} It is adequate to view the TKTs λ K ( G ) ∼ ϰ H ( G ) {\displaystyle \lambda _{K}(G)\sim \varkappa _{H}(G)} as equivalent. Since we have K λ ( i ) = ker ⁡ ( V i ) = ker ⁡ ( T π ( i ) ) = H ϰ ( π ( i ) ) = K π ~ − 1 ( ϰ ( π ( i ) ) ) , {\displaystyle K_{\lambda (i)}=\ker(V_{i})=\ker(T_{\pi (i)})=H_{\varkappa (\pi (i))}=K_{{\tilde {\pi }}^{-1}(\varkappa (\pi (i)))},} the relation between λ {\displaystyle \lambda } and ϰ {\displaystyle \varkappa } is given by λ = π ~ − 1 ∘ ϰ ∘ π {\displaystyle \lambda ={\tilde {\pi }}^{-1}\circ \varkappa \circ \pi } . Therefore, λ {\displaystyle \lambda } is another representative of the orbit ϰ S p + 1 {\displaystyle \varkappa ^{S_{p+1}}} of ϰ {\displaystyle \varkappa } under the action ( π , μ ) ↦ π ~ − 1 ∘ μ ∘ π {\displaystyle (\pi ,\mu )\mapsto {\tilde {\pi }}^{-1}\circ \mu \circ \pi } of the symmetric group S p + 1 {\displaystyle S_{p+1}} on the set of all mappings from { 1 , … , p + 1 } -> { 0 , 1 , … , p + 1 } , {\displaystyle \\{1,\ldots ,p+1\\}\to \\{0,1,\ldots ,p+1\\},} where the extension π ~ ∈ S p + 2 {\displaystyle {\tilde {\pi }}\in S_{p+2}} of the permutation π ∈ S p + 1 {\displaystyle \pi \in S_{p+1}} is defined by π ~ ( 0 ) = 0 , {\displaystyle {\tilde {\pi }}(0)=0,} and formally H 0 = G , K 0 = G . {\displaystyle H_{0}=G,K_{0}=G.} Definition. The orbit ϰ ( G ) = ϰ S p + 1 {\displaystyle \varkappa (G)=\varkappa ^{S_{p+1}}} of any representative ϰ {\displaystyle \varkappa } is an invariant of the p-group G {\displaystyle G} and is called its transfer kernel type, briefly TKT. Remark. Let # H 0 ( G ) := # { 1 ≤ i ≤ p + 1 ∣ ϰ ( i ) = 0 } {\displaystyle \\#{\mathcal {H}}_{0}(G):=\\#\\{1\leq i\leq p+1\mid \varkappa (i)=0\\}} denote the counter of total transfer kernels ker ⁡ ( T i ) = G {\displaystyle \ker(T_{i})=G} , which is an invariant of the group G {\displaystyle G} . In 1980, S. M. Chang and R. Foote[12] proved that, for any odd prime p {\displaystyle p} and for any integer 0 ≤ n ≤ p + 1 {\displaystyle 0\leq n\leq p+1} , there exist metabelian p-groups G {\displaystyle G} having abelianization G / G ′ {\displaystyle G/G'} of type ( p , p ) {\displaystyle (p,p)} such that # H 0 ( G ) = n {\displaystyle \\#{\mathcal {H}}_{0}(G)=n} . However, for p = 2 {\displaystyle p=2} , there do not exist non-abelian 2 {\displaystyle 2} -groups G {\displaystyle G} with G / G ′ ≃ ( 2 , 2 ) {\displaystyle G/G'\simeq (2,2)} , which must be metabelian of maximal class, such that # H 0 ( G ) ≥ 2 {\displaystyle \\#{\mathcal {H}}_{0}(G)\geq 2} . Only the elementary abelian 2 {\displaystyle 2} -group G = C 2 × C 2 {\displaystyle G=C_{2}\times C_{2}} has # H 0 ( G ) = 3 {\displaystyle \\#{\mathcal {H}}_{0}(G)=3} . See Figure 5. In the following concrete examples for the counters # H 0 ( G ) {\displaystyle \\#{\mathcal {H}}_{0}(G)} , and also in the remainder of this article, we use identifiers of finite p-groups in the SmallGroups Library by H. U. Besche, B. Eick and E. A. O'Brien.[13][14] For p = 3 {\displaystyle p=3} , we have * # H 0 ( G ) = 0 {\displaystyle \\#{\mathcal {H}}_{0}(G)=0} for the extra special group G = ⟨ 27 , 4 ⟩ {\displaystyle G=\langle 27,4\rangle } of exponent 9 {\displaystyle 9} with TKT ϰ = ( 1111 ) {\displaystyle \varkappa =(1111)} (Figure 6), * # H 0 ( G ) = 1 {\displaystyle \\#{\mathcal {H}}_{0}(G)=1} for the two groups G ∈ { ⟨ 243 , 6 ⟩ , ⟨ 243 , 8 ⟩ } {\displaystyle G\in \\{\langle 243,6\rangle ,\langle 243,8\rangle \\}} with TKTs ϰ ∈ { ( 0122 ) , ( 2034 ) } {\displaystyle \varkappa \in \\{(0122),(2034)\\}} (Figures 8 and 9), * # H 0 ( G ) = 2 {\displaystyle \\#{\mathcal {H}}_{0}(G)=2} for the group G = ⟨ 243 , 3 ⟩ {\displaystyle G=\langle 243,3\rangle } with TKT ϰ = ( 0043 ) {\displaystyle \varkappa =(0043)} (Figure 4 in the article on descendant trees), * # H 0 ( G ) = 3 {\displaystyle \\#{\mathcal {H}}_{0}(G)=3} for the group G = ⟨ 81 , 7 ⟩ {\displaystyle G=\langle 81,7\rangle } with TKT ϰ = ( 2000 ) {\displaystyle \varkappa =(2000)} (Figure 6), * # H 0 ( G ) = 4 {\displaystyle \\#{\mathcal {H}}_{0}(G)=4} for the extra special group G = ⟨ 27 , 3 ⟩ {\displaystyle G=\langle 27,3\rangle } of exponent 3 {\displaystyle 3} with TKT ϰ = ( 0000 ) {\displaystyle \varkappa =(0000)} (Figure 6). ## Abelianization of type (p2,p)[edit] Let G {\displaystyle G} be a p-group with abelianization G / G ′ {\displaystyle G/G'} of non-elementary abelian type ( p 2 , p ) . {\displaystyle (p^{2},p).} Then G {\displaystyle G} possesses p + 1 {\displaystyle p+1} maximal subgroups H 1 , … , H p + 1 {\displaystyle H_{1},\ldots ,H_{p+1}} of index p {\displaystyle p} and p + 1 {\displaystyle p+1} subgroups U 1 , … , U p + 1 {\displaystyle U_{1},\ldots ,U_{p+1}} of index p 2 . {\displaystyle p^{2}.} Assumption. Suppose H p + 1 = ∏ j = 1 p + 1 U j {\displaystyle H_{p+1}=\prod _{j=1}^{p+1}U_{j}} is the distinguished maximal subgroup and U p + 1 = ⋂ j = 1 p + 1 H j {\displaystyle U_{p+1}=\bigcap _{j=1}^{p+1}H_{j}} is the distinguished subgroup of index p 2 {\displaystyle p^{2}} which as the intersection of all maximal subgroups, is the Frattini subgroup Φ ( G ) {\displaystyle \Phi (G)} of G {\displaystyle G} . ### First layer[edit] For each 1 ≤ i ≤ p + 1 {\displaystyle 1\leq i\leq p+1} , let T 1 , i : G -> H i / H i ′ {\displaystyle T_{1,i}:G\to H_{i}/H_{i}'} denote the Artin transfer homomorphism. Definition. The family ϰ 1 , H , U ( G ) = ( ker ⁡ ( T 1 , i ) ) i = 1 p + 1 {\displaystyle \varkappa _{1,H,U}(G)=(\ker(T_{1,i}))_{i=1}^{p+1}} is called the first layer transfer kernel type of G {\displaystyle G} with respect to H 1 , … , H p + 1 {\displaystyle H_{1},\ldots ,H_{p+1}} and U 1 , … , U p + 1 {\displaystyle U_{1},\ldots ,U_{p+1}} , and is identified with ( ϰ 1 ( i ) ) i = 1 p + 1 {\displaystyle (\varkappa _{1}(i))_{i=1}^{p+1}} , where ϰ 1 ( i ) = { 0 ker ⁡ ( T 1 , i ) = H p + 1 , j ker ⁡ ( T 1 , i ) = U j for some 1 ≤ j ≤ p + 1. {\displaystyle \varkappa _{1}(i)={\begin{cases}0&\ker(T_{1,i})=H_{p+1},\\\j&\ker(T_{1,i})=U_{j}{\text{ for some }}1\leq j\leq p+1.\end{cases}}} Remark. Here, we observe that each first layer transfer kernel is of exponent p {\displaystyle p} with respect to G ′ {\displaystyle G'} and consequently cannot coincide with H j {\displaystyle H_{j}} for any 1 ≤ j ≤ p {\displaystyle 1\leq j\leq p} , since H j / G ′ {\displaystyle H_{j}/G'} is cyclic of order p 2 {\displaystyle p^{2}} , whereas H p + 1 / G ′ {\displaystyle H_{p+1}/G'} is bicyclic of type ( p , p ) {\displaystyle (p,p)} . ### Second layer[edit] For each 1 ≤ i ≤ p + 1 {\displaystyle 1\leq i\leq p+1} , let T 2 , i : G -> U i / U i ′ {\displaystyle T_{2,i}:G\to U_{i}/U_{i}'} be the Artin transfer homomorphism from G {\displaystyle G} to the abelianization of U i {\displaystyle U_{i}} . Definition. The family ϰ 2 , U , H ( G ) = ( ker ⁡ ( T 2 , i ) ) i = 1 p + 1 {\displaystyle \varkappa _{2,U,H}(G)=(\ker(T_{2,i}))_{i=1}^{p+1}} is called the second layer transfer kernel type of G {\displaystyle G} with respect to U 1 , … , U p + 1 {\displaystyle U_{1},\ldots ,U_{p+1}} and H 1 , … , H p + 1 {\displaystyle H_{1},\ldots ,H_{p+1}} , and is identified with ( ϰ 2 ( i ) ) i = 1 p + 1 , {\displaystyle (\varkappa _{2}(i))_{i=1}^{p+1},} where ϰ 2 ( i ) = { 0 ker ⁡ ( T 2 , i ) = G , j ker ⁡ ( T 2 , i ) = H j for some 1 ≤ j ≤ p + 1. {\displaystyle \varkappa _{2}(i)={\begin{cases}0&\ker(T_{2,i})=G,\\\j&\ker(T_{2,i})=H_{j}{\text{ for some }}1\leq j\leq p+1.\end{cases}}} ### Transfer kernel type[edit] Combining the information on the two layers, we obtain the (complete) transfer kernel type ϰ H , U ( G ) = ( ϰ 1 , H , U ( G ) ; ϰ 2 , U , H ( G ) ) {\displaystyle \varkappa _{H,U}(G)=(\varkappa _{1,H,U}(G);\varkappa _{2,U,H}(G))} of the p-group G {\displaystyle G} with respect to H 1 , … , H p + 1 {\displaystyle H_{1},\ldots ,H_{p+1}} and U 1 , … , U p + 1 {\displaystyle U_{1},\ldots ,U_{p+1}} . Remark. The distinguished subgroups H p + 1 {\displaystyle H_{p+1}} and U p + 1 = Φ ( G ) {\displaystyle U_{p+1}=\Phi (G)} are unique invariants of G {\displaystyle G} and should not be renumerated. However, independent renumerations of the remaining maximal subgroups K i = H τ ( i ) ( 1 ≤ i ≤ p ) {\displaystyle K_{i}=H_{\tau (i)}(1\leq i\leq p)} and the transfers V 1 , i = T 1 , τ ( i ) {\displaystyle V_{1,i}=T_{1,\tau (i)}} by means of a permutation τ ∈ S p {\displaystyle \tau \in S_{p}} , and of the remaining subgroups W i = U σ ( i ) ( 1 ≤ i ≤ p ) {\displaystyle W_{i}=U_{\sigma (i)}(1\leq i\leq p)} of index p 2 {\displaystyle p^{2}} and the transfers V 2 , i = T 2 , σ ( i ) {\displaystyle V_{2,i}=T_{2,\sigma (i)}} by means of a permutation σ ∈ S p {\displaystyle \sigma \in S_{p}} , give rise to new TKTs λ 1 , K , W ( G ) = ( ker ⁡ ( V 1 , i ) ) i = 1 p + 1 {\displaystyle \lambda _{1,K,W}(G)=(\ker(V_{1,i}))_{i=1}^{p+1}} with respect to K 1 , … , K p + 1 {\displaystyle K_{1},\ldots ,K_{p+1}} and W 1 , … , W p + 1 {\displaystyle W_{1},\ldots ,W_{p+1}} , identified with ( λ 1 ( i ) ) i = 1 p + 1 {\displaystyle (\lambda _{1}(i))_{i=1}^{p+1}} , where λ 1 ( i ) = { 0 ker ⁡ ( V 1 , i ) = K p + 1 , j ker ⁡ ( V 1 , i ) = W j for some 1 ≤ j ≤ p + 1 , {\displaystyle \lambda _{1}(i)={\begin{cases}0&\ker(V_{1,i})=K_{p+1},\\\j&\ker(V_{1,i})=W_{j}{\text{ for some }}1\leq j\leq p+1,\end{cases}}} and λ 2 , W , K ( G ) = ( ker ⁡ ( V 2 , i ) ) i = 1 p + 1 {\displaystyle \lambda _{2,W,K}(G)=(\ker(V_{2,i}))_{i=1}^{p+1}} with respect to W 1 , … , W p + 1 {\displaystyle W_{1},\ldots ,W_{p+1}} and K 1 , … , K p + 1 {\displaystyle K_{1},\ldots ,K_{p+1}} , identified with ( λ 2 ( i ) ) i = 1 p + 1 , {\displaystyle (\lambda _{2}(i))_{i=1}^{p+1},} where λ 2 ( i ) = { 0 ker ⁡ ( V 2 , i ) = G , j ker ⁡ ( V 2 , i ) = K j for some 1 ≤ j ≤ p + 1. {\displaystyle \lambda _{2}(i)={\begin{cases}0&\ker(V_{2,i})=G,\\\j&\ker(V_{2,i})=K_{j}{\text{ for some }}1\leq j\leq p+1.\end{cases}}} It is adequate to view the TKTs λ 1 , K , W ( G ) ∼ ϰ 1 , H , U ( G ) {\displaystyle \lambda _{1,K,W}(G)\sim \varkappa _{1,H,U}(G)} and λ 2 , W , K ( G ) ∼ ϰ 2 , U , H ( G ) {\displaystyle \lambda _{2,W,K}(G)\sim \varkappa _{2,U,H}(G)} as equivalent. Since we have W λ 1 ( i ) = ker ⁡ ( V 1 , i ) = ker ⁡ ( T 1 , τ ^ ( i ) ) = U ϰ 1 ( τ ^ ( i ) ) = W σ ~ − 1 ( ϰ 1 ( τ ^ ( i ) ) ) K λ 2 ( i ) = ker ⁡ ( V 2 , i ) = ker ⁡ ( T 2 , σ ^ ( i ) ) = H ϰ 2 ( σ ^ ( i ) ) = K τ ~ − 1 ( ϰ 2 ( σ ^ ( i ) ) ) {\displaystyle {\begin{aligned}W_{\lambda _{1}(i)}&=\ker(V_{1,i})=\ker(T_{1,{\hat {\tau }}(i)})=U_{\varkappa _{1}({\hat {\tau }}(i))}=W_{{\tilde {\sigma }}^{-1}(\varkappa _{1}({\hat {\tau }}(i)))}\\\K_{\lambda _{2}(i)}&=\ker(V_{2,i})=\ker(T_{2,{\hat {\sigma }}(i)})=H_{\varkappa _{2}({\hat {\sigma }}(i))}=K_{{\tilde {\tau }}^{-1}(\varkappa _{2}({\hat {\sigma }}(i)))}\end{aligned}}} the relations between λ 1 {\displaystyle \lambda _{1}} and ϰ 1 {\displaystyle \varkappa _{1}} , and λ 2 {\displaystyle \lambda _{2}} and ϰ 2 {\displaystyle \varkappa _{2}} , are given by λ 1 = σ ~ − 1 ∘ ϰ 1 ∘ τ ^ {\displaystyle \lambda _{1}={\tilde {\sigma }}^{-1}\circ \varkappa _{1}\circ {\hat {\tau }}} λ 2 = τ ~ − 1 ∘ ϰ 2 ∘ σ ^ {\displaystyle \lambda _{2}={\tilde {\tau }}^{-1}\circ \varkappa _{2}\circ {\hat {\sigma }}} Therefore, λ = ( λ 1 , λ 2 ) {\displaystyle \lambda =(\lambda _{1},\lambda _{2})} is another representative of the orbit ϰ S p × S p {\displaystyle \varkappa ^{S_{p}\times S_{p}}} of ϰ = ( ϰ 1 , ϰ 2 ) {\displaystyle \varkappa =(\varkappa _{1},\varkappa _{2})} under the action: ( ( σ , τ ) , ( μ 1 , μ 2 ) ) ↦ ( σ ~ − 1 ∘ μ 1 ∘ τ ^ , τ ~ − 1 ∘ μ 2 ∘ σ ^ ) {\displaystyle ((\sigma ,\tau ),(\mu _{1},\mu _{2}))\mapsto \left({\tilde {\sigma }}^{-1}\circ \mu _{1}\circ {\hat {\tau }},{\tilde {\tau }}^{-1}\circ \mu _{2}\circ {\hat {\sigma }}\right)} of the product of two symmetric groups S p × S p {\displaystyle S_{p}\times S_{p}} on the set of all pairs of mappings { 1 , … , p + 1 } -> { 0 , 1 , … , p + 1 } {\displaystyle \\{1,\ldots ,p+1\\}\to \\{0,1,\ldots ,p+1\\}} , where the extensions π ^ ∈ S p + 1 {\displaystyle {\hat {\pi }}\in S_{p+1}} and π ~ ∈ S p + 2 {\displaystyle {\tilde {\pi }}\in S_{p+2}} of a permutation π ∈ S p {\displaystyle \pi \in S_{p}} are defined by π ^ ( p + 1 ) = π ~ ( p + 1 ) = p + 1 {\displaystyle {\hat {\pi }}(p+1)={\tilde {\pi }}(p+1)=p+1} and π ~ ( 0 ) = 0 {\displaystyle {\tilde {\pi }}(0)=0} , and formally H 0 = K 0 = G , K p + 1 = H p + 1 , U 0 = W 0 = H p + 1 , {\displaystyle H_{0}=K_{0}=G,K_{p+1}=H_{p+1},U_{0}=W_{0}=H_{p+1},} and W p + 1 = U p + 1 = Φ ( G ) . {\displaystyle W_{p+1}=U_{p+1}=\Phi (G).} Definition. The orbit ϰ ( G ) = ϰ S p × S p {\displaystyle \varkappa (G)=\varkappa ^{S_{p}\times S_{p}}} of any representative ϰ = ( ϰ 1 , ϰ 2 ) {\displaystyle \varkappa =(\varkappa _{1},\varkappa _{2})} is an invariant of the p-group G {\displaystyle G} and is called its transfer kernel type, briefly TKT. ### Connections between layers[edit] The Artin transfer T 2 , i : G -> U i / U i ′ {\displaystyle T_{2,i}:G\to U_{i}/U_{i}'} is the composition T 2 , i = T ~ H j , U i ∘ T 1 , j {\displaystyle T_{2,i}={\tilde {T}}_{H_{j},U_{i}}\circ T_{1,j}} of the induced transfer T ~ H j , U i : H j / H j ′ -> U i / U i ′ {\displaystyle {\tilde {T}}_{H_{j},U_{i}}:H_{j}/H_{j}'\to U_{i}/U_{i}'} from H j {\displaystyle H_{j}} to U i {\displaystyle U_{i}} and the Artin transfer T 1 , j : G -> H j / H j ′ . {\displaystyle T_{1,j}:G\to H_{j}/H_{j}'.} There are two options regarding the intermediate subgroups * For the subgroups U 1 , … , U p {\displaystyle U_{1},\ldots ,U_{p}} only the distinguished maximal subgroup H p + 1 {\displaystyle H_{p+1}} is an intermediate subgroup. * For the Frattini subgroup U p + 1 = Φ ( G ) {\displaystyle U_{p+1}=\Phi (G)} all maximal subgroups H 1 , … , H p + 1 {\displaystyle H_{1},\ldots ,H_{p+1}} are intermediate subgroups. This causes restrictions for the transfer kernel type ϰ 2 ( G ) {\displaystyle \varkappa _{2}(G)} of the second layer, since ker ⁡ ( T 2 , i ) = ker ⁡ ( T ~ H j , U i ∘ T 1 , j ) ⊃ ker ⁡ ( T 1 , j ) , {\displaystyle \ker(T_{2,i})=\ker({\tilde {T}}_{H_{j},U_{i}}\circ T_{1,j})\supset \ker(T_{1,j}),} and thus ∀ i ∈ { 1 , … , p } : ker ⁡ ( T 2 , i ) ⊃ ker ⁡ ( T 1 , p + 1 ) . {\displaystyle \forall i\in \\{1,\ldots ,p\\}:\qquad \ker(T_{2,i})\supset \ker(T_{1,p+1}).} But even ker ⁡ ( T 2 , p + 1 ) ⊃ ⟨ ⋃ j = 1 p + 1 ker ⁡ ( T 1 , j ) ⟩ . {\displaystyle \ker(T_{2,p+1})\supset \left\langle \bigcup _{j=1}^{p+1}\ker(T_{1,j})\right\rangle .} Furthermore, when G = ⟨ x , y ⟩ {\displaystyle G=\langle x,y\rangle } with x p ∉ G ′ , y p ∈ G ′ , {\displaystyle x^{p}\notin G',y^{p}\in G',} an element x y k − 1 ( 1 ≤ k ≤ p ) {\displaystyle xy^{k-1}(1\leq k\leq p)} of order p 2 {\displaystyle p^{2}} with respect to G ′ {\displaystyle G'} , can belong to ker ⁡ ( T 2 , i ) {\displaystyle \ker(T_{2,i})} only if its p {\displaystyle p} th power is contained in ker ⁡ ( T 1 , j ) {\displaystyle \ker(T_{1,j})} , for all intermediate subgroups U i < H j < G {\displaystyle U_{i} G ~ {\displaystyle \phi :G\to {\tilde {G}}} with ker ⁡ ( ϕ ) = N . {\displaystyle \ker(\phi )=N.} Then the group G ~ = ϕ ( G ) {\displaystyle {\tilde {G}}=\phi (G)} can be viewed as the parent of the descendant G {\displaystyle G} . In the following sections, this point of view will be taken, generally for arbitrary groups, not only for finite p-groups. ### Passing through the abelianization[edit] Proposition. Suppose A {\displaystyle A} is an abelian group and ϕ : G -> A {\displaystyle \phi :G\to A} is a homomorphism. Let ω : G -> G / G ′ {\displaystyle \omega :G\to G/G'} denote the canonical projection map. Then there exists a unique homomorphism ϕ ~ : G / G ′ -> A {\displaystyle {\tilde {\phi }}:G/G'\to A} such that ϕ = ϕ ~ ∘ ω {\displaystyle \phi ={\tilde {\phi }}\circ \omega } and ker ⁡ ( ϕ ~ ) = ker ⁡ ( ϕ ) / G ′ {\displaystyle \ker({\tilde {\phi }})=\ker(\phi )/G'} (See Figure 1). Proof. This statement is a consequence of the second Corollary in the article on the induced homomorphism. Nevertheless, we give an independent proof for the present situation: the uniqueness of ϕ ~ {\displaystyle {\tilde {\phi }}} is a consequence of the condition ϕ = ϕ ~ ∘ ω , {\displaystyle \phi ={\tilde {\phi }}\circ \omega ,} which implies for any x ∈ G {\displaystyle x\in G} we have: ϕ ~ ( x G ′ ) = ϕ ~ ( ω ( x ) ) = ( ϕ ~ ∘ ω ) ( x ) = ϕ ( x ) , {\displaystyle {\tilde {\phi }}(xG')={\tilde {\phi }}(\omega (x))=({\tilde {\phi }}\circ \omega )(x)=\phi (x),} ϕ ~ {\displaystyle {\tilde {\phi }}} is a homomorphism, let x , y ∈ G {\displaystyle x,y\in G} be arbitrary, then: ϕ ~ ( x G ′ ⋅ y G ′ ) = ϕ ~ ( ( x y ) G ′ ) = ϕ ( x y ) = ϕ ( x ) ⋅ ϕ ( y ) = ϕ ~ ( x G ′ ) ⋅ ϕ ~ ( x G ′ ) ϕ ( [ x , y ] ) = ϕ ( x − 1 y − 1 x y ) = ϕ ( x − 1 ) ϕ ( y − 1 ) ϕ ( x ) ϕ ( y ) = [ ϕ ( x ) , ϕ ( y ) ] = 1 A is abelian. {\displaystyle {\begin{aligned}{\tilde {\phi }}\left(xG'\cdot yG'\right)&={\tilde {\phi }}((xy)G')=\phi (xy)=\phi (x)\cdot \phi (y)={\tilde {\phi }}(xG')\cdot {\tilde {\phi }}(xG')\\\\\phi ([x,y])&=\phi \left(x^{-1}y^{-1}xy\right)=\phi (x^{-1})\phi (y^{-1})\phi (x)\phi (y)=[\phi (x),\phi (y)]=1&&A{\text{ is abelian.}}\end{aligned}}} Thus, the commutator subgroup G ′ ⊂ ker ⁡ ( ϕ ) {\displaystyle G'\subset \ker(\phi )} , and this finally shows that the definition of ϕ ~ {\displaystyle {\tilde {\phi }}} is independent of the coset representative, x G ′ = y G ′ ⟹ y − 1 x ∈ G ′ ⊂ ker ⁡ ( ϕ ) ⟹ 1 = ϕ ( y − 1 x ) = ϕ ~ ( y − 1 x G ′ ) = ϕ ~ ( y G ′ ) − 1 ⋅ ϕ ~ ( x G ′ ) ⟹ ϕ ~ ( x G ′ ) = ϕ ~ ( y G ′ ) {\displaystyle {\begin{aligned}xG'=yG'&\Longrightarrow y^{-1}x\in G'\subset \ker(\phi )\\\&\Longrightarrow 1=\phi (y^{-1}x)={\tilde {\phi }}(y^{-1}xG')={\tilde {\phi }}(yG')^{-1}\cdot {\tilde {\phi }}(xG')\\\&\Longrightarrow {\tilde {\phi }}(xG')={\tilde {\phi }}(yG')\end{aligned}}} Figure 2: Epimorphisms and derived quotients. ### TTT singulets[edit] Proposition. Assume G , G ~ , ϕ {\displaystyle G,{\tilde {G}},\phi } are as above and H ~ = ϕ ( H ) {\displaystyle {\tilde {H}}=\phi (H)} is the image of a subgroup H . {\displaystyle H.} The commutator subgroup of H ~ {\displaystyle {\tilde {H}}} is the image of the commutator subgroup of H . {\displaystyle H.} Therefore, ϕ {\displaystyle \phi } induces a unique epimorphism ϕ ~ : H / H ′ -> H ~ / H ~ ′ {\displaystyle {\tilde {\phi }}:H/H'\to {\tilde {H}}/{\tilde {H}}'} , and thus H ~ / H ~ ′ {\displaystyle {\tilde {H}}/{\tilde {H}}'} is a quotient of H / H ′ . {\displaystyle H/H'.} Moreover, if ker ⁡ ( ϕ ) ≤ H ′ {\displaystyle \ker(\phi )\leq H'} , then the map ϕ ~ {\displaystyle {\tilde {\phi }}} is an isomorphism (See Figure 2). Proof. This claim is a consequence of the Main Theorem in the article on the induced homomorphism. Nevertheless, an independent proof is given as follows: first, the image of the commutator subgroup is ϕ ( H ′ ) = ϕ ( [ H , H ] ) = ϕ ( ⟨ [ u , v ] | u , v ∈ H ⟩ ) = ⟨ [ ϕ ( u ) , ϕ ( v ) ] ∣ u , v ∈ H ⟩ = [ ϕ ( H ) , ϕ ( H ) ] = ϕ ( H ) ′ = H ~ ′ . {\displaystyle \phi (H')=\phi ([H,H])=\phi (\langle [u,v]|u,v\in H\rangle )=\langle [\phi (u),\phi (v)]\mid u,v\in H\rangle =[\phi (H),\phi (H)]=\phi (H)'={\tilde {H}}'.} Second, the epimorphism ϕ {\displaystyle \phi } can be restricted to an epimorphism ϕ | H : H -> H ~ {\displaystyle \phi |_{H}:H\to {\tilde {H}}} . According to the previous section, the composite epimorphism ( ω H ~ ∘ ϕ | H ) : H -> H ~ / H ~ ′ {\displaystyle (\omega _{\tilde {H}}\circ \phi |_{H}):H\to {\tilde {H}}/{\tilde {H}}'} factors through H / H ′ {\displaystyle H/H'} by means of a uniquely determined epimorphism ϕ ~ : H / H ′ -> H ~ / H ~ ′ {\displaystyle {\tilde {\phi }}:H/H'\to {\tilde {H}}/{\tilde {H}}'} such that ϕ ~ ∘ ω H = ω H ~ ∘ ϕ | H {\displaystyle {\tilde {\phi }}\circ \omega _{H}=\omega _{\tilde {H}}\circ \phi |_{H}} . Consequently, we have H ~ / H ~ ′ ≃ ( H / H ′ ) / ker ⁡ ( ϕ ~ ) {\displaystyle {\tilde {H}}/{\tilde {H}}'\simeq (H/H')/\ker({\tilde {\phi }})} . Furthermore, the kernel of ϕ ~ {\displaystyle {\tilde {\phi }}} is given explicitly by ker ⁡ ( ϕ ~ ) = ( H ′ ⋅ ker ⁡ ( ϕ ) ) / H ′ {\displaystyle \ker({\tilde {\phi }})=(H'\cdot \ker(\phi ))/H'} . Finally, if ker ⁡ ( ϕ ) ≤ H ′ {\displaystyle \ker(\phi )\leq H'} , then ϕ ~ {\displaystyle {\tilde {\phi }}} is an isomorphism, since ker ⁡ ( ϕ ~ ) = H ′ / H ′ = 1 {\displaystyle \ker({\tilde {\phi }})=H'/H'=1} . Definition.[15] Due to the results in the present section, it makes sense to define a partial order on the set of abelian type invariants by putting H ~ / H ~ ′ ⪯ H / H ′ {\displaystyle {\tilde {H}}/{\tilde {H}}'\preceq H/H'} , when H ~ / H ~ ′ ≃ ( H / H ′ ) / ker ⁡ ( ϕ ~ ) {\displaystyle {\tilde {H}}/{\tilde {H}}'\simeq (H/H')/\ker({\tilde {\phi }})} , and H ~ / H ~ ′ = H / H ′ {\displaystyle {\tilde {H}}/{\tilde {H}}'=H/H'} , when H ~ / H ~ ′ ≃ H / H ′ {\displaystyle {\tilde {H}}/{\tilde {H}}'\simeq H/H'} . Figure 3: Epimorphisms and Artin transfers. ### TKT singulets[edit] Proposition. Assume G , G ~ , ϕ {\displaystyle G,{\tilde {G}},\phi } are as above and H ~ = ϕ ( H ) {\displaystyle {\tilde {H}}=\phi (H)} is the image of a subgroup of finite index n . {\displaystyle n.} Let T G , H : G -> H / H ′ {\displaystyle T_{G,H}:G\to H/H'} and T G ~ , H ~ : G ~ -> H ~ / H ~ ′ {\displaystyle T_{{\tilde {G}},{\tilde {H}}}:{\tilde {G}}\to {\tilde {H}}/{\tilde {H}}'} be Artin transfers. If ker ⁡ ( ϕ ) ≤ H {\displaystyle \ker(\phi )\leq H} , then the image of a left transversal of H {\displaystyle H} in G {\displaystyle G} is a left transversal of H ~ {\displaystyle {\tilde {H}}} in G ~ {\displaystyle {\tilde {G}}} , and ϕ ( ker ⁡ ( T G , H ) ) ⊂ ker ⁡ ( T G ~ , H ~ ) . {\displaystyle \phi (\ker(T_{G,H}))\subset \ker(T_{{\tilde {G}},{\tilde {H}}}).} Moreover, if ker ⁡ ( ϕ ) ≤ H ′ {\displaystyle \ker(\phi )\leq H'} then ϕ ( ker ⁡ ( T G , H ) ) = ker ⁡ ( T G ~ , H ~ ) {\displaystyle \phi (\ker(T_{G,H}))=\ker(T_{{\tilde {G}},{\tilde {H}}})} (See Figure 3). Proof. Let ( g 1 , … , g n ) {\displaystyle (g_{1},\ldots ,g_{n})} be a left transversal of H {\displaystyle H} in G {\displaystyle G} . Then we have a disjoint union: G = ⨆ i = 1 n g i H . {\displaystyle G=\bigsqcup _{i=1}^{n}g_{i}H.} Consider the image of this disjoint union, which is not necessarily disjoint, ϕ ( G ) = ⋃ i = 1 n ϕ ( g i ) ϕ ( H ) , {\displaystyle \phi (G)=\bigcup _{i=1}^{n}\phi (g_{i})\phi (H),} and let j , k ∈ { 1 , … , n } . {\displaystyle j,k\in \\{1,\ldots ,n\\}.} We have: ϕ ( g j ) ϕ ( H ) = ϕ ( g k ) ϕ ( H ) ⟺ ϕ ( H ) = ϕ ( g j ) − 1 ϕ ( g k ) ϕ ( H ) = ϕ ( g j − 1 g k ) ϕ ( H ) ⟺ ϕ ( g j − 1 g k ) = ϕ ( h ) for some h ∈ H ⟺ ϕ ( h − 1 g j − 1 g k ) = 1 ⟺ h − 1 g j − 1 g k ∈ ker ⁡ ( ϕ ) ⊂ H ⟺ g j − 1 g k ∈ H ⟺ j = k {\displaystyle {\begin{aligned}\phi (g_{j})\phi (H)=\phi (g_{k})\phi (H)&\Longleftrightarrow \phi (H)=\phi (g_{j})^{-1}\phi (g_{k})\phi (H)=\phi (g_{j}^{-1}g_{k})\phi (H)\\\&\Longleftrightarrow \phi (g_{j}^{-1}g_{k})=\phi (h)&&{\text{for some }}h\in H\\\&\Longleftrightarrow \phi (h^{-1}g_{j}^{-1}g_{k})=1\\\&\Longleftrightarrow h^{-1}g_{j}^{-1}g_{k}\in \ker(\phi )\subset H\\\&\Longleftrightarrow g_{j}^{-1}g_{k}\in H\\\&\Longleftrightarrow j=k\\\\\end{aligned}}} Let ϕ ~ : H / H ′ -> H ~ / H ~ ′ {\displaystyle {\tilde {\phi }}:H/H'\to {\tilde {H}}/{\tilde {H}}'} be the epimorphism from the previous proposition. We have: ϕ ~ ( T G , H ( x ) ) = ϕ ~ ( ∏ i = 1 n g π x ( i ) − 1 x g i ⋅ H ′ ) = ∏ i = 1 n ϕ ( g π x ( i ) ) − 1 ϕ ( x ) ϕ ( g i ) ⋅ ϕ ( H ′ ) . {\displaystyle {\tilde {\phi }}(T_{G,H}(x))={\tilde {\phi }}\left(\prod _{i=1}^{n}g_{\pi _{x}(i)}^{-1}xg_{i}\cdot H'\right)=\prod _{i=1}^{n}\phi \left(g_{\pi _{x}(i)}\right)^{-1}\phi (x)\phi (g_{i})\cdot \phi (H').} Since ϕ ( H ′ ) = ϕ ( H ) ′ = H ~ ′ {\displaystyle \phi (H')=\phi (H)'={\tilde {H}}'} , the right hand side equals T G ~ , H ~ ( ϕ ( x ) ) {\displaystyle T_{{\tilde {G}},{\tilde {H}}}(\phi (x))} , if ( ϕ ( g 1 ) , … , ϕ ( g n ) ) {\displaystyle (\phi (g_{1}),\ldots ,\phi (g_{n}))} is a left transversal of H ~ {\displaystyle {\tilde {H}}} in G ~ {\displaystyle {\tilde {G}}} , which is true when ker ⁡ ( ϕ ) ⊂ H . {\displaystyle \ker(\phi )\subset H.} Therefore, ϕ ~ ∘ T G , H = T G ~ , H ~ ∘ ϕ . {\displaystyle {\tilde {\phi }}\circ T_{G,H}=T_{{\tilde {G}},{\tilde {H}}}\circ \phi .} Consequently, ker ⁡ ( ϕ ) ⊂ H {\displaystyle \ker(\phi )\subset H} implies the inclusion ϕ ( ker ⁡ ( T G , H ) ) ⊂ ker ⁡ ( T G ~ , H ~ ) . {\displaystyle \phi (\ker(T_{G,H}))\subset \ker(T_{{\tilde {G}},{\tilde {H}}}).} Finally, if ker ⁡ ( ϕ ) ⊂ H ′ {\displaystyle \ker(\phi )\subset H'} , then by the previous proposition ϕ ~ {\displaystyle {\tilde {\phi }}} is an isomorphism. Using its inverse we get T G , H = ϕ ~ − 1 ∘ T G ~ , H ~ ∘ ϕ {\displaystyle T_{G,H}={\tilde {\phi }}^{-1}\circ T_{{\tilde {G}},{\tilde {H}}}\circ \phi } , which proves ϕ − 1 ( ker ⁡ ( T G ~ , H ~ ) ) ⊂ ker ⁡ ( T G , H ) . {\displaystyle \phi ^{-1}\left(\ker(T_{{\tilde {G}},{\tilde {H}}})\right)\subset \ker(T_{G,H}).} Combining the inclusions we have: { ϕ ( ker ⁡ ( T G , H ) ) ⊂ ker ⁡ ( T G ~ , H ~ ) ϕ − 1 ( ker ⁡ ( T G ~ , H ~ ) ) ⊂ ker ⁡ ( T G , H ) ⟹ { ϕ ( ker ⁡ ( T G , H ) ) ⊂ ker ⁡ ( T G ~ , H ~ ) ϕ ( ϕ − 1 ( ker ⁡ ( T G ~ , H ~ ) ) ) ⊂ ϕ ( ker ⁡ ( T G , H ) ) ⟹ ϕ ( ϕ − 1 ( ker ⁡ ( T G ~ , H ~ ) ) ) ⊂ ϕ ( ker ⁡ ( T G , H ) ) ⊂ ker ⁡ ( T G ~ , H ~ ) ⟹ ker ⁡ ( T G ~ , H ~ ) ⊂ ϕ ( ker ⁡ ( T G , H ) ) ⊂ ker ⁡ ( T G ~ , H ~ ) ⟹ ϕ ( ker ⁡ ( T G , H ) ) = ker ⁡ ( T G ~ , H ~ ) {\displaystyle {\begin{aligned}{\begin{cases}\phi (\ker(T_{G,H}))\subset \ker(T_{{\tilde {G}},{\tilde {H}}})\\\\\phi ^{-1}\left(\ker(T_{{\tilde {G}},{\tilde {H}}})\right)\subset \ker(T_{G,H})\end{cases}}&\Longrightarrow {\begin{cases}\phi (\ker(T_{G,H}))\subset \ker(T_{{\tilde {G}},{\tilde {H}}})\\\\\phi \left(\phi ^{-1}\left(\ker(T_{{\tilde {G}},{\tilde {H}}})\right)\right)\subset \phi (\ker(T_{G,H}))\end{cases}}\\\\[8pt]&\Longrightarrow \phi \left(\phi ^{-1}\left(\ker(T_{{\tilde {G}},{\tilde {H}}})\right)\right)\subset \phi (\ker(T_{G,H}))\subset \ker(T_{{\tilde {G}},{\tilde {H}}})\\\\[8pt]&\Longrightarrow \ker(T_{{\tilde {G}},{\tilde {H}}})\subset \phi (\ker(T_{G,H}))\subset \ker(T_{{\tilde {G}},{\tilde {H}}})\\\\[8pt]&\Longrightarrow \phi (\ker(T_{G,H}))=\ker(T_{{\tilde {G}},{\tilde {H}}})\end{aligned}}} Definition.[15] In view of the results in the present section, we are able to define a partial order of transfer kernels by setting ker ⁡ ( T G , H ) ⪯ ker ⁡ ( T G ~ , H ~ ) {\displaystyle \ker(T_{G,H})\preceq \ker(T_{{\tilde {G}},{\tilde {H}}})} , when ϕ ( ker ⁡ ( T G , H ) ) ⊂ ker ⁡ ( T G ~ , H ~ ) . {\displaystyle \phi (\ker(T_{G,H}))\subset \ker(T_{{\tilde {G}},{\tilde {H}}}).} ### TTT and TKT multiplets[edit] Assume G , G ~ , ϕ {\displaystyle G,{\tilde {G}},\phi } are as above and that G / G ′ {\displaystyle G/G'} and G ~ / G ~ ′ {\displaystyle {\tilde {G}}/{\tilde {G}}'} are isomorphic and finite. Let ( H i ) i ∈ I {\displaystyle (H_{i})_{i\in I}} denote the family of all subgroups containing G ′ {\displaystyle G'} (making it a finite family of normal subgroups). For each i ∈ I {\displaystyle i\in I} let: H i ~ := ϕ ( H i ) T i := T G , H i : G -> H i / H i ′ T i ~ := T G ~ , H i ~ : G ~ -> H i ~ / H i ~ ′ {\displaystyle {\begin{aligned}{\tilde {H_{i}}}&:=\phi (H_{i})\\\T_{i}&:=T_{G,H_{i}}:G\to H_{i}/H_{i}'\\\\{\tilde {T_{i}}}&:=T_{{\tilde {G}},{\tilde {H_{i}}}}:{\tilde {G}}\to {\tilde {H_{i}}}/{\tilde {H_{i}}}'\end{aligned}}} Take J {\displaystyle J} be any non-empty subset of I {\displaystyle I} . Then it is convenient to define ϰ H ( G ) = ( ker ⁡ ( T j ) ) j ∈ J {\displaystyle \varkappa _{H}(G)=(\ker(T_{j}))_{j\in J}} , called the (partial) transfer kernel type (TKT) of G {\displaystyle G} with respect to ( H j ) j ∈ J {\displaystyle (H_{j})_{j\in J}} , and τ H ( G ) = ( H j / H j ′ ) j ∈ J , {\displaystyle \tau _{H}(G)=(H_{j}/H_{j}')_{j\in J},} called the (partial) transfer target type (TTT) of G {\displaystyle G} with respect to ( H j ) j ∈ J {\displaystyle (H_{j})_{j\in J}} . Due to the rules for singulets, established in the preceding two sections, these multiplets of TTTs and TKTs obey the following fundamental inheritance laws: Inheritance Law I. If ker ⁡ ( ϕ ) ≤ ∩ j ∈ J H j {\displaystyle \ker(\phi )\leq \cap _{j\in J}H_{j}} , then τ H ~ ( G ~ ) ⪯ τ H ( G ) {\displaystyle \tau _{\tilde {H}}({\tilde {G}})\preceq \tau _{H}(G)} , in the sense that H j ~ / H j ~ ′ ⪯ H j / H j ′ {\displaystyle {\tilde {H_{j}}}/{\tilde {H_{j}}}'\preceq H_{j}/H_{j}'} , for each j ∈ J {\displaystyle j\in J} , and ϰ H ( G ) ⪯ ϰ H ~ ( G ~ ) {\displaystyle \varkappa _{H}(G)\preceq \varkappa _{\tilde {H}}({\tilde {G}})} , in the sense that ker ⁡ ( T j ) ⪯ ker ⁡ ( T j ~ ) {\displaystyle \ker(T_{j})\preceq \ker({\tilde {T_{j}}})} , for each j ∈ J {\displaystyle j\in J} . Inheritance Law II. If ker ⁡ ( ϕ ) ≤ ∩ j ∈ J H j ′ {\displaystyle \ker(\phi )\leq \cap _{j\in J}H_{j}'} , then τ H ~ ( G ~ ) = τ H ( G ) {\displaystyle \tau _{\tilde {H}}({\tilde {G}})=\tau _{H}(G)} , in the sense that H j ~ / H j ~ ′ = H j / H j ′ {\displaystyle {\tilde {H_{j}}}/{\tilde {H_{j}}}'=H_{j}/H_{j}'} , for each j ∈ J {\displaystyle j\in J} , and ϰ H ( G ) = ϰ H ~ ( G ~ ) {\displaystyle \varkappa _{H}(G)=\varkappa _{\tilde {H}}({\tilde {G}})} , in the sense that ker ⁡ ( T j ) = ker ⁡ ( T j ~ ) {\displaystyle \ker(T_{j})=\ker({\tilde {T_{j}}})} , for each j ∈ J {\displaystyle j\in J} . ### Inherited automorphisms[edit] A further inheritance property does not immediately concern Artin transfers but will prove to be useful in applications to descendant trees. Inheritance Law III. Assume G , G ~ , ϕ {\displaystyle G,{\tilde {G}},\phi } are as above and σ ∈ A u t ( G ) . {\displaystyle \sigma \in \mathrm {Aut} (G).} If σ ( ker ⁡ ( ϕ ) ) ⊂ ker ⁡ ( ϕ ) {\displaystyle \sigma (\ker(\phi ))\subset \ker(\phi )} then there exists a unique epimorphism σ ~ : G ~ -> G ~ {\displaystyle {\tilde {\sigma }}:{\tilde {G}}\to {\tilde {G}}} such that ϕ ∘ σ = σ ~ ∘ ϕ {\displaystyle \phi \circ \sigma ={\tilde {\sigma }}\circ \phi } . If σ ( ker ⁡ ( ϕ ) ) = ker ⁡ ( ϕ ) , {\displaystyle \sigma (\ker(\phi ))=\ker(\phi ),} then σ ~ ∈ A u t ( G ~ ) . {\displaystyle {\tilde {\sigma }}\in \mathrm {Aut} ({\tilde {G}}).} Proof. Using the isomorphism G ~ = ϕ ( G ) ≃ G / ker ⁡ ( ϕ ) {\displaystyle {\tilde {G}}=\phi (G)\simeq G/\ker(\phi )} we define: { σ ~ : G ~ -> G ~ σ ~ ( g ker ⁡ ( ϕ ) ) := σ ( g ) ker ⁡ ( ϕ ) {\displaystyle {\begin{cases}{\tilde {\sigma }}:{\tilde {G}}\to {\tilde {G}}\\\\{\tilde {\sigma }}(g\ker(\phi )):=\sigma (g)\ker(\phi )\end{cases}}} First we show this map is well-defined: g ker ⁡ ( ϕ ) = h ker ⁡ ( ϕ ) ⟹ h − 1 g ∈ ker ⁡ ( ϕ ) ⟹ σ ( h − 1 g ) ∈ σ ( ker ⁡ ( ϕ ) ) ⟹ σ ( h − 1 g ) ∈ ker ⁡ ( ϕ ) σ ( ker ⁡ ( ϕ ) ) ⊂ ker ⁡ ( ϕ ) ⟹ σ ( h − 1 ) σ ( g ) ∈ ker ⁡ ( ϕ ) ⟹ σ ( g ) ker ⁡ ( ϕ ) = σ ( h ) ker ⁡ ( ϕ ) {\displaystyle {\begin{aligned}g\ker(\phi )=h\ker(\phi )&\Longrightarrow h^{-1}g\in \ker(\phi )\\\&\Longrightarrow \sigma (h^{-1}g)\in \sigma (\ker(\phi ))\\\&\Longrightarrow \sigma (h^{-1}g)\in \ker(\phi )&&\sigma (\ker(\phi ))\subset \ker(\phi )\\\&\Longrightarrow \sigma (h^{-1})\sigma (g)\in \ker(\phi )\\\&\Longrightarrow \sigma (g)\ker(\phi )=\sigma (h)\ker(\phi )\end{aligned}}} The fact that σ ~ {\displaystyle {\tilde {\sigma }}} is surjective, a homomorphism and satisfies ϕ ∘ σ = σ ~ ∘ ϕ {\displaystyle \phi \circ \sigma ={\tilde {\sigma }}\circ \phi } are easily verified. And if σ ( ker ⁡ ( ϕ ) ) = ker ⁡ ( ϕ ) {\displaystyle \sigma (\ker(\phi ))=\ker(\phi )} , then injectivity of σ ~ {\displaystyle {\tilde {\sigma }}} is a consequence of σ ~ ( g ker ⁡ ( ϕ ) ) = ker ⁡ ( ϕ ) ⟹ σ ( g ) ker ⁡ ( ϕ ) = ker ⁡ ( ϕ ) ⟹ σ ( g ) ∈ ker ⁡ ( ϕ ) ⟹ σ − 1 ( σ ( g ) ) ∈ σ − 1 ( ker ⁡ ( ϕ ) ) ⟹ g ∈ σ − 1 ( ker ⁡ ( ϕ ) ) ⟹ g ∈ ker ⁡ ( ϕ ) σ − 1 ( ker ⁡ ( ϕ ) ) ⊂ ker ⁡ ( ϕ ) ⟹ g ker ⁡ ( ϕ ) = ker ⁡ ( ϕ ) {\displaystyle {\begin{aligned}{\tilde {\sigma }}(g\ker(\phi ))=\ker(\phi )&\Longrightarrow \sigma (g)\ker(\phi )=\ker(\phi )\\\&\Longrightarrow \sigma (g)\in \ker(\phi )\\\&\Longrightarrow \sigma ^{-1}(\sigma (g))\in \sigma ^{-1}(\ker(\phi ))\\\&\Longrightarrow g\in \sigma ^{-1}(\ker(\phi ))\\\&\Longrightarrow g\in \ker(\phi )&&\sigma ^{-1}(\ker(\phi ))\subset \ker(\phi )\\\&\Longrightarrow g\ker(\phi )=\ker(\phi )\end{aligned}}} Let ω : G -> G / G ′ {\displaystyle \omega :G\to G/G'} be the canonical projection then there exists a unique induced automorphism σ ¯ ∈ A u t ( G / G ′ ) {\displaystyle {\bar {\sigma }}\in \mathrm {Aut} (G/G')} such that ω ∘ σ = σ ¯ ∘ ω {\displaystyle \omega \circ \sigma ={\bar {\sigma }}\circ \omega } , that is, ∀ g ∈ G : σ ¯ ( g G ′ ) = σ ¯ ( ω ( g ) ) = ω ( σ ( g ) ) = σ ( g ) G ′ , {\displaystyle \forall g\in G:\qquad {\bar {\sigma }}(gG')={\bar {\sigma }}(\omega (g))=\omega (\sigma (g))=\sigma (g)G',} The reason for the injectivity of σ ¯ {\displaystyle {\bar {\sigma }}} is that σ ( g ) G ′ = σ ¯ ( g G ′ ) = G ′ ⇒ σ ( g ) ∈ G ′ ⇒ g = σ − 1 ( σ ( g ) ) ∈ G ′ , {\displaystyle \sigma (g)G'={\bar {\sigma }}(gG')=G'\Rightarrow \sigma (g)\in G'\Rightarrow g=\sigma ^{-1}(\sigma (g))\in G',} since G ′ {\displaystyle G'} is a characteristic subgroup of G {\displaystyle G} . Definition. G {\displaystyle G} is called a σ−group, if there exists σ ∈ A u t ( G ) {\displaystyle \sigma \in \mathrm {Aut} (G)} such that the induced automorphism acts like the inversion on G / G ′ {\displaystyle G/G'} , that is for all g ∈ G : σ ( g ) G ′ = σ ¯ ( g G ′ ) = g − 1 G ′ ⟺ σ ( g ) g ∈ G ′ . {\displaystyle g\in G:\qquad \sigma (g)G'={\bar {\sigma }}(gG')=g^{-1}G'\Longleftrightarrow \sigma (g)g\in G'.} The Inheritance Law III asserts that, if G {\displaystyle G} is a σ−group and σ ( ker ⁡ ( ϕ ) ) = ker ⁡ ( ϕ ) {\displaystyle \sigma (\ker(\phi ))=\ker(\phi )} , then G ~ {\displaystyle {\tilde {G}}} is also a σ−group, the required automorphism being σ ~ {\displaystyle {\tilde {\sigma }}} . This can be seen by applying the epimorphism ϕ {\displaystyle \phi } to the equation σ ( g ) G ′ = σ ¯ ( g G ′ ) = g − 1 G ′ {\displaystyle \sigma (g)G'={\bar {\sigma }}(gG')=g^{-1}G'} which yields ∀ x = ϕ ( g ) ∈ ϕ ( G ) = G ~ : σ ~ ( x ) G ~ ′ = σ ~ ( ϕ ( g ) ) G ~ ′ = ϕ ( σ ( g ) ) ϕ ( G ′ ) = ϕ ( g − 1 ) ϕ ( G ′ ) = ϕ ( g ) − 1 G ~ ′ = x − 1 G ~ ′ . {\displaystyle \forall x=\phi (g)\in \phi (G)={\tilde {G}}:\qquad {\tilde {\sigma }}(x){\tilde {G}}'={\tilde {\sigma }}(\phi (g)){\tilde {G}}'=\phi (\sigma (g))\phi (G')=\phi (g^{-1})\phi (G')=\phi (g)^{-1}{\tilde {G}}'=x^{-1}{\tilde {G}}'.} ## Stabilization criteria[edit] In this section, the results concerning the inheritance of TTTs and TKTs from quotients in the previous section are applied to the simplest case, which is characterized by the following Assumption. The parent π ( G ) {\displaystyle \pi (G)} of a group G {\displaystyle G} is the quotient π ( G ) = G / N {\displaystyle \pi (G)=G/N} of G {\displaystyle G} by the last non-trivial term N = γ c ( G ) ◃ G {\displaystyle N=\gamma _{c}(G)\triangleleft G} of the lower central series of G {\displaystyle G} , where c {\displaystyle c} denotes the nilpotency class of G {\displaystyle G} . The corresponding epimorphism π {\displaystyle \pi } from G {\displaystyle G} onto π ( G ) = G / γ c ( G ) {\displaystyle \pi (G)=G/\gamma _{c}(G)} is the canonical projection, whose kernel is given by ker ⁡ ( π ) = γ c ( G ) {\displaystyle \ker(\pi )=\gamma _{c}(G)} . Under this assumption, the kernels and targets of Artin transfers turn out to be compatible with parent-descendant relations between finite p-groups. Compatibility criterion. Let p {\displaystyle p} be a prime number. Suppose that G {\displaystyle G} is a non-abelian finite p-group of nilpotency class c = c l ( G ) ≥ 2 {\displaystyle c=\mathrm {cl} (G)\geq 2} . Then the TTT and the TKT of G {\displaystyle G} and of its parent π ( G ) {\displaystyle \pi (G)} are comparable in the sense that τ ( π ( G ) ) ⪯ τ ( G ) {\displaystyle \tau (\pi (G))\preceq \tau (G)} and ϰ ( G ) ⪯ ϰ ( π ( G ) ) {\displaystyle \varkappa (G)\preceq \varkappa (\pi (G))} . The simple reason for this fact is that, for any subgroup G ′ ≤ H ≤ G {\displaystyle G'\leq H\leq G} , we have ker ⁡ ( π ) = γ c ( G ) ≤ γ 2 ( G ) = G ′ ≤ H {\displaystyle \ker(\pi )=\gamma _{c}(G)\leq \gamma _{2}(G)=G'\leq H} , since c ≥ 2 {\displaystyle c\geq 2} . For the remaining part of this section, the investigated groups are supposed to be finite metabelian p-groups G {\displaystyle G} with elementary abelianization G / G ′ {\displaystyle G/G'} of rank 2 {\displaystyle 2} , that is of type ( p , p ) {\displaystyle (p,p)} . Partial stabilization for maximal class. A metabelian p-group G {\displaystyle G} of coclass c c ( G ) = 1 {\displaystyle \mathrm {cc} (G)=1} and of nilpotency class c = c l ( G ) ≥ 3 {\displaystyle c=\mathrm {cl} (G)\geq 3} shares the last p {\displaystyle p} components of the TTT τ ( G ) {\displaystyle \tau (G)} and of the TKT ϰ ( G ) {\displaystyle \varkappa (G)} with its parent π ( G ) {\displaystyle \pi (G)} . More explicitly, for odd primes p ≥ 3 {\displaystyle p\geq 3} , we have τ ( G ) i = ( p , p ) {\displaystyle \tau (G)_{i}=(p,p)} and ϰ ( G ) i = 0 {\displaystyle \varkappa (G)_{i}=0} for 2 ≤ i ≤ p + 1 {\displaystyle 2\leq i\leq p+1} . [16] This criterion is due to the fact that c ≥ 3 {\displaystyle c\geq 3} implies ker ⁡ ( π ) = γ c ( G ) ≤ γ 3 ( G ) = H i ′ {\displaystyle \ker(\pi )=\gamma _{c}(G)\leq \gamma _{3}(G)=H_{i}'} , [17] for the last p {\displaystyle p} maximal subgroups H 2 , … , H p + 1 {\displaystyle H_{2},\ldots ,H_{p+1}} of G {\displaystyle G} . The condition c ≥ 3 {\displaystyle c\geq 3} is indeed necessary for the partial stabilization criterion. For odd primes p ≥ 3 {\displaystyle p\geq 3} , the extra special p {\displaystyle p} -group G = G 0 3 ( 0 , 1 ) {\displaystyle G=G_{0}^{3}(0,1)} of order p 3 {\displaystyle p^{3}} and exponent p 2 {\displaystyle p^{2}} has nilpotency class c = 2 {\displaystyle c=2} only, and the last p {\displaystyle p} components of its TKT ϰ = ( 1 p + 1 ) {\displaystyle \varkappa =(1^{p+1})} are strictly smaller than the corresponding components of the TKT ϰ = ( 0 p + 1 ) {\displaystyle \varkappa =(0^{p+1})} of its parent π ( G ) {\displaystyle \pi (G)} which is the elementary abelian p {\displaystyle p} -group of type ( p , p ) {\displaystyle (p,p)} . [16] For p = 2 {\displaystyle p=2} , both extra special 2 {\displaystyle 2} -groups of coclass 1 {\displaystyle 1} and class c = 2 {\displaystyle c=2} , the ordinary quaternion group G = G 0 3 ( 0 , 1 ) {\displaystyle G=G_{0}^{3}(0,1)} with TKT ϰ = ( 123 ) {\displaystyle \varkappa =(123)} and the dihedral group G = G 0 3 ( 0 , 0 ) {\displaystyle G=G_{0}^{3}(0,0)} with TKT ϰ = ( 023 ) {\displaystyle \varkappa =(023)} , have strictly smaller last two components of their TKTs than their common parent π ( G ) = C 2 × C 2 {\displaystyle \pi (G)=C_{2}\times C_{2}} with TKT ϰ = ( 000 ) {\displaystyle \varkappa =(000)} . Total stabilization for maximal class and positive defect. A metabelian p-group G {\displaystyle G} of coclass c c ( G ) = 1 {\displaystyle \mathrm {cc} (G)=1} and of nilpotency class c = m − 1 = c l ( G ) ≥ 4 {\displaystyle c=m-1=\mathrm {cl} (G)\geq 4} , that is, with index of nilpotency m ≥ 5 {\displaystyle m\geq 5} , shares all p + 1 {\displaystyle p+1} components of the TTT τ ( G ) {\displaystyle \tau (G)} and of the TKT ϰ ( G ) {\displaystyle \varkappa (G)} with its parent π ( G ) {\displaystyle \pi (G)} , provided it has positive defect of commutativity k = k ( G ) ≥ 1 {\displaystyle k=k(G)\geq 1} . [11] Note that k ≥ 1 {\displaystyle k\geq 1} implies p ≥ 3 {\displaystyle p\geq 3} , and we have ϰ ( G ) i = 0 {\displaystyle \varkappa (G)_{i}=0} for all 1 ≤ i ≤ p + 1 {\displaystyle 1\leq i\leq p+1} . [16] This statement can be seen by observing that the conditions m ≥ 5 {\displaystyle m\geq 5} and k ≥ 1 {\displaystyle k\geq 1} imply ker ⁡ ( π ) = γ m − 1 ( G ) ≤ γ m − k ( G ) ≤ H i ′ {\displaystyle \ker(\pi )=\gamma _{m-1}(G)\leq \gamma _{m-k}(G)\leq H_{i}'} , [17] for all the p + 1 {\displaystyle p+1} maximal subgroups H 1 , … , H p + 1 {\displaystyle H_{1},\ldots ,H_{p+1}} of G {\displaystyle G} . The condition k ≥ 1 {\displaystyle k\geq 1} is indeed necessary for total stabilization. To see this it suffices to consider the first component of the TKT only. For each nilpotency class c ≥ 4 {\displaystyle c\geq 4} , there exist (at least) two groups G = G 0 c + 1 ( 0 , 1 ) {\displaystyle G=G_{0}^{c+1}(0,1)} with TKT ϰ = ( 10 p ) {\displaystyle \varkappa =(10^{p})} and G = G 0 c + 1 ( 1 , 0 ) {\displaystyle G=G_{0}^{c+1}(1,0)} with TKT ϰ = ( 20 p ) {\displaystyle \varkappa =(20^{p})} , both with defect k = 0 {\displaystyle k=0} , where the first component of their TKT is strictly smaller than the first component of the TKT ϰ = ( 0 p + 1 ) {\displaystyle \varkappa =(0^{p+1})} of their common parent π ( G ) = G 0 c ( 0 , 0 ) {\displaystyle \pi (G)=G_{0}^{c}(0,0)} . Partial stabilization for non-maximal class. Let p = 3 {\displaystyle p=3} be fixed. A metabelian 3-group G {\displaystyle G} with abelianization G / G ′ ≃ ( 3 , 3 ) {\displaystyle G/G'\simeq (3,3)} , coclass c c ( G ) ≥ 2 {\displaystyle \mathrm {cc} (G)\geq 2} and nilpotency class c = c l ( G ) ≥ 4 {\displaystyle c=\mathrm {cl} (G)\geq 4} shares the last two (among the four) components of the TTT τ ( G ) {\displaystyle \tau (G)} and of the TKT ϰ ( G ) {\displaystyle \varkappa (G)} with its parent π ( G ) {\displaystyle \pi (G)} . This criterion is justified by the following consideration. If c ≥ 4 {\displaystyle c\geq 4} , then ker ⁡ ( π ) = γ c ( G ) ≤ γ 4 ( G ) ≤ H i ′ {\displaystyle \ker(\pi )=\gamma _{c}(G)\leq \gamma _{4}(G)\leq H_{i}'} [17] for the last two maximal subgroups H 3 , H 4 {\displaystyle H_{3},H_{4}} of G {\displaystyle G} . The condition c ≥ 4 {\displaystyle c\geq 4} is indeed unavoidable for partial stabilization, since there exist several 3 {\displaystyle 3} -groups of class c = 3 {\displaystyle c=3} , for instance those with SmallGroups identifiers G ∈ { ⟨ 243 , 3 ⟩ , ⟨ 243 , 6 ⟩ , ⟨ 243 , 8 ⟩ } {\displaystyle G\in \\{\langle 243,3\rangle ,\langle 243,6\rangle ,\langle 243,8\rangle \\}} , such that the last two components of their TKTs ϰ ∈ { ( 0043 ) , ( 0122 ) , ( 2034 ) } {\displaystyle \varkappa \in \\{(0043),(0122),(2034)\\}} are strictly smaller than the last two components of the TKT ϰ = ( 0000 ) {\displaystyle \varkappa =(0000)} of their common parent π ( G ) = G 0 3 ( 0 , 0 ) {\displaystyle \pi (G)=G_{0}^{3}(0,0)} . Total stabilization for non-maximal class and cyclic centre. Again, let p = 3 {\displaystyle p=3} be fixed. A metabelian 3-group G {\displaystyle G} with abelianization G / G ′ ≃ ( 3 , 3 ) {\displaystyle G/G'\simeq (3,3)} , coclass c c ( G ) ≥ 2 {\displaystyle \mathrm {cc} (G)\geq 2} , nilpotency class c = c l ( G ) ≥ 4 {\displaystyle c=\mathrm {cl} (G)\geq 4} and cyclic centre ζ 1 ( G ) {\displaystyle \zeta _{1}(G)} shares all four components of the TTT τ ( G ) {\displaystyle \tau (G)} and of the TKT ϰ ( G ) {\displaystyle \varkappa (G)} with its parent π ( G ) {\displaystyle \pi (G)} . The reason is that, due to the cyclic centre, we have ker ⁡ ( π ) = γ c ( G ) = ζ 1 ( G ) ≤ H i ′ {\displaystyle \ker(\pi )=\gamma _{c}(G)=\zeta _{1}(G)\leq H_{i}'} [17] for all four maximal subgroups H 1 , … , H 4 {\displaystyle H_{1},\ldots ,H_{4}} of G {\displaystyle G} . The condition of a cyclic centre is indeed necessary for total stabilization, since for a group with bicyclic centre there occur two possibilities. Either γ c ( G ) = ζ 1 ( G ) {\displaystyle \gamma _{c}(G)=\zeta _{1}(G)} is also bicyclic, whence γ c ( G ) {\displaystyle \gamma _{c}(G)} is never contained in H 2 ′ {\displaystyle H_{2}'} , or γ c ( G ) < ζ 1 ( G ) {\displaystyle \gamma _{c}(G)<\zeta _{1}(G)} is cyclic but is never contained in H 1 ′ {\displaystyle H_{1}'} . Summarizing, we can say that the last four criteria underpin the fact that Artin transfers provide a marvellous tool for classifying finite p-groups. In the following sections, it will be shown how these ideas can be applied for endowing descendant trees with additional structure, and for searching particular groups in descendant trees by looking for patterns defined by the kernels and targets of Artin transfers. These strategies of pattern recognition are useful in pure group theory and in algebraic number theory. Figure 4: Endowing a descendant tree with information on Artin transfers. ## Structured descendant trees (SDTs)[edit] This section uses the terminology of descendant trees in the theory of finite p-groups. In Figure 4, a descendant tree with modest complexity is selected exemplarily to demonstrate how Artin transfers provide additional structure for each vertex of the tree. More precisely, the underlying prime is p = 3 {\displaystyle p=3} , and the chosen descendant tree is actually a coclass tree having a unique infinite mainline, branches of depth 3 {\displaystyle 3} , and strict periodicity of length 2 {\displaystyle 2} setting in with branch B ( 7 ) {\displaystyle {\mathcal {B}}(7)} . The initial pre-period consists of branches B ( 5 ) {\displaystyle {\mathcal {B}}(5)} and B ( 6 ) {\displaystyle {\mathcal {B}}(6)} with exceptional structure. Branches B ( 7 ) {\displaystyle {\mathcal {B}}(7)} and B ( 8 ) {\displaystyle {\mathcal {B}}(8)} form the primitive period such that B ( j ) ≃ B ( 7 ) {\displaystyle {\mathcal {B}}(j)\simeq {\mathcal {B}}(7)} , for odd j ≥ 9 {\displaystyle j\geq 9} , and B ( j ) ≃ B ( 8 ) {\displaystyle {\mathcal {B}}(j)\simeq {\mathcal {B}}(8)} , for even j ≥ 10 {\displaystyle j\geq 10} . The root of the tree is the metabelian 3 {\displaystyle 3} -group with identifier R = ⟨ 243 , 6 ⟩ {\displaystyle R=\langle 243,6\rangle } , that is, a group of order | R | = 3 5 = 243 {\displaystyle |R|=3^{5}=243} and with counting number 6 {\displaystyle 6} . This root is not coclass settled, whence its entire descendant tree T ( R ) {\displaystyle {\mathcal {T}}(R)} is of considerably higher complexity than the coclass- 2 {\displaystyle 2} subtree T 2 ( R ) {\displaystyle {\mathcal {T}}^{2}(R)} , whose first six branches are drawn in the diagram of Figure 4. The additional structure can be viewed as a sort of coordinate system in which the tree is embedded. The horizontal abscissa is labelled with the transfer kernel type (TKT) ϰ {\displaystyle \varkappa } , and the vertical ordinate is labelled with a single component τ ( 1 ) {\displaystyle \tau (1)} of the transfer target type (TTT). The vertices of the tree are drawn in such a manner that members of periodic infinite sequences form a vertical column sharing a common TKT. On the other hand, metabelian groups of a fixed order, represented by vertices of depth at most 1 {\displaystyle 1} , form a horizontal row sharing a common first component of the TTT. (To discourage any incorrect interpretations, we explicitly point out that the first component of the TTT of non-metabelian groups or metabelian groups, represented by vertices of depth 2 {\displaystyle 2} , is usually smaller than expected, due to stabilization phenomena!) The TTT of all groups in this tree represented by a big full disk, which indicates a bicyclic centre of type ( 3 , 3 ) {\displaystyle (3,3)} , is given by τ = [ A ( 3 , c ) , ( 3 , 3 , 3 ) , ( 9 , 3 ) , ( 9 , 3 ) ] {\displaystyle \tau =[A(3,c),(3,3,3),(9,3),(9,3)]} with varying first component τ ( 1 ) = A ( 3 , c ) {\displaystyle \tau (1)=A(3,c)} , the nearly homocyclic abelian 3 {\displaystyle 3} -group of order 3 c {\displaystyle 3^{c}} , and fixed further components τ ( 2 ) = ( 3 , 3 , 3 ) = ^ ( 1 3 ) {\displaystyle \tau (2)=(3,3,3){\hat {=}}(1^{3})} and τ ( 3 ) = τ ( 4 ) = ( 9 , 3 ) = ^ ( 21 ) {\displaystyle \tau (3)=\tau (4)=(9,3){\hat {=}}(21)} , where the abelian type invariants are either written as orders of cyclic components or as their 3 {\displaystyle 3} -logarithms with exponents indicating iteration. (The latter notation is employed in Figure 4.) Since the coclass of all groups in this tree is 2 {\displaystyle 2} , the connection between the order 3 n {\displaystyle 3^{n}} and the nilpotency class is given by c = n − 2 {\displaystyle c=n-2} . ## Pattern recognition[edit] For searching a particular group in a descendant tree by looking for patterns defined by the kernels and targets of Artin transfers, it is frequently adequate to reduce the number of vertices in the branches of a dense tree with high complexity by sifting groups with desired special properties, for example * filtering the σ {\displaystyle \sigma } -groups, * eliminating a set of certain transfer kernel types, * cancelling all non-metabelian groups (indicated by small contour squares in Fig. 4), * removing metabelian groups with cyclic centre (denoted by small full disks in Fig. 4), * cutting off vertices whose distance from the mainline (depth) exceeds some lower bound, * combining several different sifting criteria. The result of such a sieving procedure is called a pruned descendant tree with respect to the desired set of properties. However, in any case, it should be avoided that the main line of a coclass tree is eliminated, since the result would be a disconnected infinite set of finite graphs instead of a tree. For example, it is neither recommended to eliminate all σ {\displaystyle \sigma } -groups in Figure 4 nor to eliminate all groups with TKT ϰ = ( 0122 ) {\displaystyle \varkappa =(0122)} . In Figure 4, the big double contour rectangle surrounds the pruned coclass tree T ∗ 2 ( R ) {\displaystyle {\mathcal {T}}_{\ast }^{2}(R)} , where the numerous vertices with TKT ϰ = ( 2122 ) {\displaystyle \varkappa =(2122)} are completely eliminated. This would, for instance, be useful for searching a σ {\displaystyle \sigma } -group with TKT ϰ = ( 1122 ) {\displaystyle \varkappa =(1122)} and first component τ ( 1 ) = ( 43 ) {\displaystyle \tau (1)=(43)} of the TTT. In this case, the search result would even be a unique group. We expand this idea further in the following detailed discussion of an important example. ### Historical example[edit] The oldest example of searching for a finite p-group by the strategy of pattern recognition via Artin transfers goes back to 1934, when A. Scholz and O. Taussky [18] tried to determine the Galois group G = G 3 ∞ ( K ) = G a l ( F 3 ∞ ( K ) | K ) {\displaystyle G=\mathrm {G} _{3}^{\infty }(K)=\mathrm {Gal} (\mathrm {F} _{3}^{\infty }(K)|K)} of the Hilbert 3 {\displaystyle 3} -class field tower, that is the maximal unramified pro- 3 {\displaystyle 3} extension F 3 ∞ ( K ) {\displaystyle \mathrm {F} _{3}^{\infty }(K)} , of the complex quadratic number field K = Q ( − 9748 ) . {\displaystyle K=\mathbb {Q} ({\sqrt {-9748}}).} They actually succeeded in finding the maximal metabelian quotient Q = G / G ″ = G 3 2 ( K ) = G a l ( F 3 2 ( K ) | K ) {\displaystyle Q=G/G''=\mathrm {G} _{3}^{2}(K)=\mathrm {Gal} (\mathrm {F} _{3}^{2}(K)|K)} of G {\displaystyle G} , that is the Galois group of the second Hilbert 3 {\displaystyle 3} -class field F 3 2 ( K ) {\displaystyle \mathrm {F} _{3}^{2}(K)} of K {\displaystyle K} . However, it needed 78 {\displaystyle 78} years until M. R. Bush and D. C. Mayer, in 2012, provided the first rigorous proof [15] that the (potentially infinite) 3 {\displaystyle 3} -tower group G = G 3 ∞ ( K ) {\displaystyle G=\mathrm {G} _{3}^{\infty }(K)} coincides with the finite 3 {\displaystyle 3} -group G 3 3 ( K ) = G a l ( F 3 3 ( K ) | K ) {\displaystyle \mathrm {G} _{3}^{3}(K)=\mathrm {Gal} (\mathrm {F} _{3}^{3}(K)|K)} of derived length d l ( G ) = 3 {\displaystyle \mathrm {dl} (G)=3} , and thus the 3 {\displaystyle 3} -tower of K {\displaystyle K} has exactly three stages, stopping at the third Hilbert 3 {\displaystyle 3} -class field F 3 3 ( K ) {\displaystyle \mathrm {F} _{3}^{3}(K)} of K {\displaystyle K} . Table 1: Possible quotients Pc of the 3-tower group G of K [15] c | order of Pc | SmallGroups identifier of Pc | TKT ϰ {\displaystyle \varkappa } of Pc | TTT τ {\displaystyle \tau } of Pc | ν | μ | descendant numbers of Pc | | | | | | | 1 {\displaystyle 1} | 9 {\displaystyle 9} | ⟨ 9 , 2 ⟩ {\displaystyle \langle 9,2\rangle } | ( 0000 ) {\displaystyle (0000)} | [ ( 1 ) ( 1 ) ( 1 ) ( 1 ) ] {\displaystyle [(1)(1)(1)(1)]} | 3 {\displaystyle 3} | 3 {\displaystyle 3} | 3 / 2 ; 3 / 3 ; 1 / 1 {\displaystyle 3/2;3/3;1/1} 2 {\displaystyle 2} | 27 {\displaystyle 27} | ⟨ 27 , 3 ⟩ {\displaystyle \langle 27,3\rangle } | ( 0000 ) {\displaystyle (0000)} | [ ( 1 2 ) ( 1 2 ) ( 1 2 ) ( 1 2 ) ] {\displaystyle [(1^{2})(1^{2})(1^{2})(1^{2})]} | 2 {\displaystyle 2} | 4 {\displaystyle 4} | 4 / 1 ; 7 / 5 {\displaystyle 4/1;7/5} 3 {\displaystyle 3} | 243 {\displaystyle 243} | ⟨ 243 , 8 ⟩ {\displaystyle \langle 243,8\rangle } | ( 2034 ) {\displaystyle (2034)} | [ ( 21 ) ( 21 ) ( 21 ) ( 21 ) ] {\displaystyle [(21)(21)(21)(21)]} | 1 {\displaystyle 1} | 3 {\displaystyle 3} | 4 / 4 {\displaystyle 4/4} 4 {\displaystyle 4} | 729 {\displaystyle 729} | ⟨ 729 , 54 ⟩ {\displaystyle \langle 729,54\rangle } | ( 2034 ) {\displaystyle (2034)} | [ ( 21 ) ( 2 2 ) ( 21 ) ( 21 ) ] {\displaystyle [(21)(2^{2})(21)(21)]} | 2 {\displaystyle 2} | 4 {\displaystyle 4} | 8 / 3 ; 6 / 3 {\displaystyle 8/3;6/3} 5 {\displaystyle 5} | 2187 {\displaystyle 2187} | ⟨ 2187 , 302 ⟩ {\displaystyle \langle 2187,302\rangle } | ( 2334 ) {\displaystyle (2334)} | [ ( 21 ) ( 32 ) ( 21 ) ( 21 ) ] {\displaystyle [(21)(32)(21)(21)]} | 0 {\displaystyle 0} | 3 {\displaystyle 3} | 0 / 0 {\displaystyle 0/0} 5 {\displaystyle 5} | 2187 {\displaystyle 2187} | ⟨ 2187 , 306 ⟩ {\displaystyle \langle 2187,306\rangle } | ( 2434 ) {\displaystyle (2434)} | [ ( 21 ) ( 32 ) ( 21 ) ( 21 ) ] {\displaystyle [(21)(32)(21)(21)]} | 0 {\displaystyle 0} | 3 {\displaystyle 3} | 0 / 0 {\displaystyle 0/0} 5 {\displaystyle 5} | 2187 {\displaystyle 2187} | ⟨ 2187 , 303 ⟩ {\displaystyle \langle 2187,303\rangle } | ( 2034 ) {\displaystyle (2034)} | [ ( 21 ) ( 32 ) ( 21 ) ( 21 ) ] {\displaystyle [(21)(32)(21)(21)]} | 1 {\displaystyle 1} | 4 {\displaystyle 4} | 5 / 2 {\displaystyle 5/2} 5 {\displaystyle 5} | 6561 {\displaystyle 6561} | ⟨ 729 , 54 ⟩ − # 2 ; 2 {\displaystyle \langle 729,54\rangle -\\#2;2} | ( 2334 ) {\displaystyle (2334)} | [ ( 21 ) ( 32 ) ( 21 ) ( 21 ) ] {\displaystyle [(21)(32)(21)(21)]} | 0 {\displaystyle 0} | 2 {\displaystyle 2} | 0 / 0 {\displaystyle 0/0} 5 {\displaystyle 5} | 6561 {\displaystyle 6561} | ⟨ 729 , 54 ⟩ − # 2 ; 6 {\displaystyle \langle 729,54\rangle -\\#2;6} | ( 2434 ) {\displaystyle (2434)} | [ ( 21 ) ( 32 ) ( 21 ) ( 21 ) ] {\displaystyle [(21)(32)(21)(21)]} | 0 {\displaystyle 0} | 2 {\displaystyle 2} | 0 / 0 {\displaystyle 0/0} 5 {\displaystyle 5} | 6561 {\displaystyle 6561} | ⟨ 729 , 54 ⟩ − # 2 ; 3 {\displaystyle \langle 729,54\rangle -\\#2;3} | ( 2034 ) {\displaystyle (2034)} | [ ( 21 ) ( 32 ) ( 21 ) ( 21 ) ] {\displaystyle [(21)(32)(21)(21)]} | 1 {\displaystyle 1} | 3 {\displaystyle 3} | 4 / 4 {\displaystyle 4/4} 6 {\displaystyle 6} | 6561 {\displaystyle 6561} | ⟨ 2187 , 303 ⟩ − # 1 ; 1 {\displaystyle \langle 2187,303\rangle -\\#1;1} | ( 2034 ) {\displaystyle (2034)} | [ ( 21 ) ( 3 2 ) ( 21 ) ( 21 ) ] {\displaystyle [(21)(3^{2})(21)(21)]} | 1 {\displaystyle 1} | 4 {\displaystyle 4} | 7 / 3 {\displaystyle 7/3} 6 {\displaystyle 6} | 19683 {\displaystyle 19683} | ⟨ 729 , 54 ⟩ − # 2 ; 3 − # 1 ; 1 {\displaystyle \langle 729,54\rangle -\\#2;3-\\#1;1} | ( 2034 ) {\displaystyle (2034)} | [ ( 21 ) ( 3 2 ) ( 21 ) ( 21 ) ] {\displaystyle [(21)(3^{2})(21)(21)]} | 2 {\displaystyle 2} | 4 {\displaystyle 4} | 8 / 3 ; 6 / 3 {\displaystyle 8/3;6/3} The search is performed with the aid of the p-group generation algorithm by M. F. Newman [19] and E. A. O'Brien. [20] For the initialization of the algorithm, two basic invariants must be determined. Firstly, the generator rank d {\displaystyle d} of the p-groups to be constructed. Here, we have p = 3 {\displaystyle p=3} and d = r 3 ( K ) = d ( C l 3 ( K ) ) {\displaystyle d=r_{3}(K)=d(\mathrm {Cl} _{3}(K))} is given by the 3 {\displaystyle 3} -class rank of the quadratic field K {\displaystyle K} . Secondly, the abelian type invariants of the 3 {\displaystyle 3} -class group C l 3 ( K ) ≃ ( 1 2 ) {\displaystyle \mathrm {Cl} _{3}(K)\simeq (1^{2})} of K {\displaystyle K} . These two invariants indicate the root of the descendant tree which will be constructed successively. Although the p-group generation algorithm is designed to use the parent-descendant definition by means of the lower exponent-p central series, it can be fitted to the definition with the aid of the usual lower central series. In the case of an elementary abelian p-group as root, the difference is not very big. So we have to start with the elementary abelian 3 {\displaystyle 3} -group of rank two, which has the SmallGroups identifier ⟨ 9 , 2 ⟩ {\displaystyle \langle 9,2\rangle } , and to construct the descendant tree T ( ⟨ 9 , 2 ⟩ ) {\displaystyle {\mathcal {T}}(\langle 9,2\rangle )} . We do that by iterating the p-group generation algorithm, taking suitable capable descendants of the previous root as the next root, always executing an increment of the nilpotency class by a unit. As explained at the beginning of the section Pattern recognition, we must prune the descendant tree with respect to the invariants TKT and TTT of the 3 {\displaystyle 3} -tower group G {\displaystyle G} , which are determined by the arithmetic of the field K {\displaystyle K} as ϰ ∈ { ( 2334 ) , ( 2434 ) } {\displaystyle \varkappa \in \\{(2334),(2434)\\}} (exactly two fixed points and no transposition) and τ = [ ( 21 ) ( 32 ) ( 21 ) ( 21 ) ] {\displaystyle \tau =[(21)(32)(21)(21)]} . Further, any quotient of G {\displaystyle G} must be a σ {\displaystyle \sigma } -group, enforced by number theoretic requirements for the quadratic field K {\displaystyle K} . The root ⟨ 9 , 2 ⟩ {\displaystyle \langle 9,2\rangle } has only a single capable descendant ⟨ 27 , 3 ⟩ {\displaystyle \langle 27,3\rangle } of type ( 1 2 ) {\displaystyle (1^{2})} . In terms of the nilpotency class, ⟨ 9 , 2 ⟩ {\displaystyle \langle 9,2\rangle } is the class- 1 {\displaystyle 1} quotient G / γ 2 ( G ) {\displaystyle G/\gamma _{2}(G)} of G {\displaystyle G} and ⟨ 27 , 3 ⟩ {\displaystyle \langle 27,3\rangle } is the class- 2 {\displaystyle 2} quotient G / γ 3 ( G ) {\displaystyle G/\gamma _{3}(G)} of G {\displaystyle G} . Since the latter has nuclear rank two, there occurs a bifurcation T ( ⟨ 27 , 3 ⟩ ) = T 1 ( ⟨ 27 , 3 ⟩ ) ⊔ T 2 ( ⟨ 27 , 3 ⟩ ) {\displaystyle {\mathcal {T}}(\langle 27,3\rangle )={\mathcal {T}}^{1}(\langle 27,3\rangle )\sqcup {\mathcal {T}}^{2}(\langle 27,3\rangle )} , where the former component T 1 ( ⟨ 27 , 3 ⟩ ) {\displaystyle {\mathcal {T}}^{1}(\langle 27,3\rangle )} can be eliminated by the stabilization criterion ϰ = ( ∗ 000 ) {\displaystyle \varkappa =(\ast 000)} for the TKT of all 3 {\displaystyle 3} -groups of maximal class. Due to the inheritance property of TKTs, only the single capable descendant ⟨ 243 , 8 ⟩ {\displaystyle \langle 243,8\rangle } qualifies as the class- 3 {\displaystyle 3} quotient G / γ 4 ( G ) {\displaystyle G/\gamma _{4}(G)} of G {\displaystyle G} . There is only a single capable σ {\displaystyle \sigma } -group ⟨ 729 , 54 ⟩ {\displaystyle \langle 729,54\rangle } among the descendants of ⟨ 243 , 8 ⟩ {\displaystyle \langle 243,8\rangle } . It is the class- 4 {\displaystyle 4} quotient G / γ 5 ( G ) {\displaystyle G/\gamma _{5}(G)} of G {\displaystyle G} and has nuclear rank two. This causes the essential bifurcation T ( ⟨ 729 , 54 ⟩ ) = T 2 ( ⟨ 729 , 54 ⟩ ) ⊔ T 3 ( ⟨ 729 , 54 ⟩ ) {\displaystyle {\mathcal {T}}(\langle 729,54\rangle )={\mathcal {T}}^{2}(\langle 729,54\rangle )\sqcup {\mathcal {T}}^{3}(\langle 729,54\rangle )} in two subtrees belonging to different coclass graphs G ( 3 , 2 ) {\displaystyle {\mathcal {G}}(3,2)} and G ( 3 , 3 ) {\displaystyle {\mathcal {G}}(3,3)} . The former contains the metabelian quotient Q = G / G ″ {\displaystyle Q=G/G''} of G {\displaystyle G} with two possibilities Q ∈ { ⟨ 2187 , 302 ⟩ , ⟨ 2187 , 306 ⟩ } {\displaystyle Q\in \\{\langle 2187,302\rangle ,\langle 2187,306\rangle \\}} , which are not balanced with relation rank r = 3 > 2 = d {\displaystyle r=3>2=d} bigger than the generator rank. The latter consists entirely of non-metabelian groups and yields the desired 3 {\displaystyle 3} -tower group G {\displaystyle G} as one among the two Schur σ {\displaystyle \sigma } -groups ⟨ 729 , 54 ⟩ − # 2 ; 2 {\displaystyle \langle 729,54\rangle -\\#2;2} and ⟨ 729 , 54 ⟩ − # 2 ; 6 {\displaystyle \langle 729,54\rangle -\\#2;6} with r = 2 = d {\displaystyle r=2=d} . Finally the termination criterion is reached at the capable vertices ⟨ 2187 , 303 ⟩ − # 1 ; 1 ∈ G ( 3 , 2 ) {\displaystyle \langle 2187,303\rangle -\\#1;1\in {\mathcal {G}}(3,2)} and ⟨ 729 , 54 ⟩ − # 2 ; 3 − # 1 ; 1 ∈ G ( 3 , 3 ) {\displaystyle \langle 729,54\rangle -\\#2;3-\\#1;1\in {\mathcal {G}}(3,3)} , since the TTT τ = [ ( 21 ) ( 3 2 ) ( 21 ) ( 21 ) ] > [ ( 21 ) ( 32 ) ( 21 ) ( 21 ) ] {\displaystyle \tau =[(21)(3^{2})(21)(21)]>[(21)(32)(21)(21)]} is too big and will even increase further, never returning to [ ( 21 ) ( 32 ) ( 21 ) ( 21 ) ] {\displaystyle [(21)(32)(21)(21)]} . The complete search process is visualized in Table 1, where, for each of the possible successive p-quotients P c = G / γ c + 1 ( G ) {\displaystyle P_{c}=G/\gamma _{c+1}(G)} of the 3 {\displaystyle 3} -tower group G = G 3 ∞ ( K ) {\displaystyle G=\mathrm {G} _{3}^{\infty }(K)} of K = Q ( − 9748 ) {\displaystyle K=\mathbb {Q} ({\sqrt {-9748}})} , the nilpotency class is denoted by c = c l ( P c ) {\displaystyle c=\mathrm {cl} (P_{c})} , the nuclear rank by ν = ν ( P c ) {\displaystyle \nu =\nu (P_{c})} , and the p-multiplicator rank by μ = μ ( P c ) {\displaystyle \mu =\mu (P_{c})} . ## Commutator calculus[edit] This section shows exemplarily how commutator calculus can be used for determining the kernels and targets of Artin transfers explicitly. As a concrete example we take the metabelian 3 {\displaystyle 3} -groups with bicyclic centre, which are represented by big full disks as vertices, of the coclass tree diagram in Figure 4. They form ten periodic infinite sequences, four, resp. six, for even, resp. odd, nilpotency class c {\displaystyle c} , and can be characterized with the aid of a parametrized polycyclic power-commutator presentation: 1 G c , n ( z , w ) = ⟨ x , y , s 2 , t 3 , s 3 , … , s c ∣ x 3 = s c w , y 3 = s 3 2 s 4 s c z , s j 3 = s j + 2 2 s j + 3 for 2 ≤ j ≤ c − 3 , s c − 2 3 = s c 2 , t 3 3 = 1 , s 2 = [ y , x ] , t 3 = [ s 2 , y ] , s j = [ s j − 1 , x ] for 3 ≤ j ≤ c ⟩ , {\displaystyle {\begin{aligned}G^{c,n}(z,w)=&\langle x,y,s_{2},t_{3},s_{3},\ldots ,s_{c}\mid {}\\\&x^{3}=s_{c}^{w},\ y^{3}=s_{3}^{2}s_{4}s_{c}^{z},\ s_{j}^{3}=s_{j+2}^{2}s_{j+3}{\text{ for }}2\leq j\leq c-3,\ s_{c-2}^{3}=s_{c}^{2},\ t_{3}^{3}=1,\\\&s_{2}=[y,x],\ t_{3}=[s_{2},y],\ s_{j}=[s_{j-1},x]{\text{ for }}3\leq j\leq c\rangle ,\end{aligned}}} where c ≥ 5 {\displaystyle c\geq 5} is the nilpotency class, 3 n {\displaystyle 3^{n}} with n = c + 2 {\displaystyle n=c+2} is the order, and 0 ≤ w ≤ 1 , − 1 ≤ z ≤ 1 {\displaystyle 0\leq w\leq 1,-1\leq z\leq 1} are parameters. The transfer target type (TTT) of the group G = G c , n ( z , w ) {\displaystyle G=G^{c,n}(z,w)} depends only on the nilpotency class c {\displaystyle c} , is independent of the parameters w , z {\displaystyle w,z} , and is given uniformly by τ = [ A ( 3 , c ) , ( 3 , 3 , 3 ) , ( 9 , 3 ) , ( 9 , 3 ) ] {\displaystyle \tau =[A(3,c),(3,3,3),(9,3),(9,3)]} . This phenomenon is called a polarization, more precisely a uni-polarization,[11] at the first component. The transfer kernel type (TKT) of the group G = G c , n ( z , w ) {\displaystyle G=G^{c,n}(z,w)} is independent of the nilpotency class c {\displaystyle c} , but depends on the parameters w , z {\displaystyle w,z} , and is given by c.18, ϰ = ( 0122 ) {\displaystyle \varkappa =(0122)} , for w = z = 0 {\displaystyle w=z=0} (a mainline group), H.4, ϰ = ( 2122 ) {\displaystyle \varkappa =(2122)} , for w = 0 , z = ± 1 {\displaystyle w=0,z=\pm 1} (two capable groups), E.6, ϰ = ( 1122 ) {\displaystyle \varkappa =(1122)} , for w = 1 , z = 0 {\displaystyle w=1,z=0} (a terminal group), and E.14, ϰ ∈ { ( 4122 ) , ( 3122 ) } {\displaystyle \varkappa \in \\{(4122),(3122)\\}} , for w = 1 , z = ± 1 {\displaystyle w=1,z=\pm 1} (two terminal groups). For even nilpotency class, the two groups of types H.4 and E.14, which differ in the sign of the parameter z {\displaystyle z} only, are isomorphic. These statements can be deduced by means of the following considerations. As a preparation, it is useful to compile a list of some commutator relations, starting with those given in the presentation, [ a , x ] = 1 {\displaystyle [a,x]=1} for a ∈ { s c , t 3 } {\displaystyle a\in \\{s_{c},t_{3}\\}} and [ a , y ] = 1 {\displaystyle [a,y]=1} for a ∈ { s 3 , … , s c , t 3 } {\displaystyle a\in \\{s_{3},\ldots ,s_{c},t_{3}\\}} , which shows that the bicyclic centre is given by ζ 1 ( G ) = ⟨ s c , t 3 ⟩ {\displaystyle \zeta _{1}(G)=\langle s_{c},t_{3}\rangle } . By means of the right product rule [ a , x y ] = [ a , y ] ⋅ [ a , x ] ⋅ [ [ a , x ] , y ] {\displaystyle [a,xy]=[a,y]\cdot [a,x]\cdot [[a,x],y]} and the right power rule [ a , y 2 ] = [ a , y ] 1 + y {\displaystyle [a,y^{2}]=[a,y]^{1+y}} , we obtain [ s 2 , x y ] = s 3 t 3 {\displaystyle [s_{2},xy]=s_{3}t_{3}} , [ s 2 , x y 2 ] = s 3 t 3 2 {\displaystyle [s_{2},xy^{2}]=s_{3}t_{3}^{2}} , and [ s j , x y ] = [ s j , x y 2 ] = [ s j , x ] = s j + 1 {\displaystyle [s_{j},xy]=[s_{j},xy^{2}]=[s_{j},x]=s_{j+1}} , for j ≥ 3 {\displaystyle j\geq 3} . The maximal subgroups of G {\displaystyle G} are taken in a similar way as in the section on the computational implementation, namely H 1 = ⟨ y , G ′ ⟩ H 2 = ⟨ x , G ′ ⟩ H 3 = ⟨ x y , G ′ ⟩ H 4 = ⟨ x y 2 , G ′ ⟩ {\displaystyle {\begin{aligned}H_{1}&=\langle y,G'\rangle \\\H_{2}&=\langle x,G'\rangle \\\H_{3}&=\langle xy,G'\rangle \\\H_{4}&=\langle xy^{2},G'\rangle \\\\\end{aligned}}} Their derived subgroups are crucial for the behavior of the Artin transfers. By making use of the general formula H i ′ = ( G ′ ) h i − 1 {\displaystyle H_{i}'=(G')^{h_{i}-1}} , where H i = ⟨ h i , G ′ ⟩ {\displaystyle H_{i}=\langle h_{i},G'\rangle } , and where we know that G ′ = ⟨ s 2 , t 3 , s 3 , … , s c ⟩ {\displaystyle G'=\langle s_{2},t_{3},s_{3},\ldots ,s_{c}\rangle } in the present situation, it follows that H 1 ′ = ⟨ s 2 y − 1 ⟩ = ⟨ t 3 ⟩ H 2 ′ = ⟨ s 2 x − 1 , … , s c − 1 x − 1 ⟩ = ⟨ s 3 , … , s c ⟩ H 3 ′ = ⟨ s 2 x y − 1 , … , s c − 1 x y − 1 ⟩ = ⟨ s 3 t 3 , s 4 , … , s c ⟩ H 4 ′ = ⟨ s 2 x y 2 − 1 , … , s c − 1 x y 2 − 1 ⟩ = ⟨ s 3 t 3 2 , s 4 , … , s c ⟩ {\displaystyle {\begin{aligned}H_{1}'&=\left\langle s_{2}^{y-1}\right\rangle =\left\langle t_{3}\right\rangle \\\H_{2}'&=\left\langle s_{2}^{x-1},\ldots ,s_{c-1}^{x-1}\right\rangle =\left\langle s_{3},\ldots ,s_{c}\right\rangle \\\H_{3}'&=\left\langle s_{2}^{xy-1},\ldots ,s_{c-1}^{xy-1}\right\rangle =\left\langle s_{3}t_{3},s_{4},\ldots ,s_{c}\right\rangle \\\H_{4}'&=\left\langle s_{2}^{xy^{2}-1},\ldots ,s_{c-1}^{xy^{2}-1}\right\rangle =\left\langle s_{3}t_{3}^{2},s_{4},\ldots ,s_{c}\right\rangle \end{aligned}}} Note that H 1 {\displaystyle H_{1}} is not far from being abelian, since H 1 ′ = ⟨ t 3 ⟩ {\displaystyle H_{1}'=\langle t_{3}\rangle } is contained in the centre ζ 1 ( G ) = ⟨ s c , t 3 ⟩ {\displaystyle \zeta _{1}(G)=\langle s_{c},t_{3}\rangle } . As the first main result, we are now in the position to determine the abelian type invariants of the derived quotients: H 1 / H 1 ′ = ⟨ y , s 2 , … , s c ⟩ H 1 ′ / H 1 ′ ≃ A ( 3 , c ) , {\displaystyle H_{1}/H_{1}'=\langle y,s_{2},\ldots ,s_{c}\rangle H_{1}'/H_{1}'\simeq A(3,c),} the unique quotient which grows with increasing nilpotency class c {\displaystyle c} , since o r d ( y ) = o r d ( s 2 ) = 3 m {\displaystyle \mathrm {ord} (y)=\mathrm {ord} (s_{2})=3^{m}} for even c = 2 m {\displaystyle c=2m} and o r d ( y ) = 3 m + 1 , o r d ( s 2 ) = 3 m {\displaystyle \mathrm {ord} (y)=3^{m+1},\mathrm {ord} (s_{2})=3^{m}} for odd c = 2 m + 1 {\displaystyle c=2m+1} , H 2 / H 2 ′ = ⟨ x , s 2 , t 3 ⟩ H 2 ′ / H 2 ′ ≃ ( 3 , 3 , 3 ) H 3 / H 3 ′ = ⟨ x y , s 2 , t 3 ⟩ H 3 ′ / H 3 ′ ≃ ( 9 , 3 ) H 4 / H 4 ′ = ⟨ x y 2 , s 2 , t 3 ⟩ H 4 ′ / H 4 ′ ≃ ( 9 , 3 ) {\displaystyle {\begin{aligned}H_{2}/H_{2}'&=\langle x,s_{2},t_{3}\rangle H_{2}'/H_{2}'\simeq (3,3,3)\\\H_{3}/H_{3}'&=\langle xy,s_{2},t_{3}\rangle H_{3}'/H_{3}'\simeq (9,3)\\\H_{4}/H_{4}'&=\langle xy^{2},s_{2},t_{3}\rangle H_{4}'/H_{4}'\simeq (9,3)\end{aligned}}} since generally o r d ( s 2 ) = o r d ( t 3 ) = 3 {\displaystyle \mathrm {ord} (s_{2})=\mathrm {ord} (t_{3})=3} , but o r d ( x ) = 3 {\displaystyle \mathrm {ord} (x)=3} for H 2 {\displaystyle H_{2}} , whereas o r d ( x y ) = o r d ( x y 2 ) = 9 {\displaystyle \mathrm {ord} (xy)=\mathrm {ord} (xy^{2})=9} for H 3 {\displaystyle H_{3}} and H 4 {\displaystyle H_{4}} . Now we come to the kernels of the Artin transfer homomorphisms T i : G -> H i / H i ′ {\displaystyle T_{i}:G\to H_{i}/H_{i}'} . It suffices to investigate the induced transfers T ~ i : G / G ′ -> H i / H i ′ {\displaystyle {\tilde {T}}_{i}:G/G'\to H_{i}/H_{i}'} and to begin by finding expressions for the images T ~ i ( g G ′ ) {\displaystyle {\tilde {T}}_{i}(gG')} of elements g G ′ ∈ G / G ′ {\displaystyle gG'\in G/G'} , which can be expressed in the form g ≡ x j y ℓ ( mod G ′ ) , j , ℓ ∈ { − 1 , 0 , 1 } . {\displaystyle g\equiv x^{j}y^{\ell }{\pmod {G'}},\qquad j,\ell \in \\{-1,0,1\\}.} First, we exploit outer transfers as much as possible: x ∉ H 1 ⇒ T ~ 1 ( x G ′ ) = x 3 H 1 ′ = s c w H 1 ′ y ∉ H 2 ⇒ T ~ 2 ( y G ′ ) = y 3 H 2 ′ = s 3 2 s 4 s c z H 2 ′ = 1 ⋅ H 2 ′ x , y ∉ H 3 , H 4 ⇒ { T ~ i ( x G ′ ) = x 3 H i ′ = s c w H i ′ = 1 ⋅ H i ′ T ~ i ( y G ′ ) = y 3 H i ′ = s 3 2 s 4 s c z H i ′ = s 3 2 H i ′ 3 ≤ i ≤ 4 {\displaystyle {\begin{aligned}x\notin H_{1}&\Rightarrow {\tilde {T}}_{1}(xG')=x^{3}H_{1}'=s_{c}^{w}H_{1}'\\\y\notin H_{2}&\Rightarrow {\tilde {T}}_{2}(yG')=y^{3}H_{2}'=s_{3}^{2}s_{4}s_{c}^{z}H_{2}'=1\cdot H_{2}'\\\x,y\notin H_{3},H_{4}&\Rightarrow {\begin{cases}{\tilde {T}}_{i}(xG')=x^{3}H_{i}'=s_{c}^{w}H_{i}'=1\cdot H_{i}'\\\\{\tilde {T}}_{i}(yG')=y^{3}H_{i}'=s_{3}^{2}s_{4}s_{c}^{z}H_{i}'=s_{3}^{2}H_{i}'\end{cases}}&&3\leq i\leq 4\end{aligned}}} Next, we treat the unavoidable inner transfers, which are more intricate. For this purpose, we use the polynomial identity X 2 + X + 1 = ( X − 1 ) 2 + 3 ( X − 1 ) + 3 {\displaystyle X^{2}+X+1=(X-1)^{2}+3(X-1)+3} to obtain: y ∈ H 1 ⇒ T ~ 1 ( y G ′ ) = y 1 + x + x 2 H 1 ′ = y 3 + 3 ( x − 1 ) + ( x − 1 ) 2 H 1 ′ = y 3 ⋅ [ y , x ] 3 ⋅ [ [ y , x ] , x ] H 1 ′ = s 3 2 s 4 s c z s 2 3 s 3 H 1 ′ = s 2 3 s 3 3 s 4 s c z H 1 ′ = s c z H 1 ′ x ∈ H 2 ⇒ T ~ 2 ( x G ′ ) = x 1 + y + y 2 H 2 ′ = x 3 + 3 ( y − 1 ) + ( y − 1 ) 2 H 2 ′ = x 3 ⋅ [ x , y ] 3 ⋅ [ [ x , y ] , y ] H 2 ′ = s c w s 2 − 3 t 3 − 1 H 2 ′ = t 3 − 1 H 2 ′ {\displaystyle {\begin{aligned}y\in H_{1}&\Rightarrow {\tilde {T}}_{1}(yG')=y^{1+x+x^{2}}H_{1}'=y^{3+3(x-1)+(x-1)^{2}}H_{1}'=y^{3}\cdot [y,x]^{3}\cdot [[y,x],x]H_{1}'=s_{3}^{2}s_{4}s_{c}^{z}s_{2}^{3}s_{3}H_{1}'=s_{2}^{3}s_{3}^{3}s_{4}s_{c}^{z}H_{1}'=s_{c}^{z}H_{1}'\\\x\in H_{2}&\Rightarrow {\tilde {T}}_{2}(xG')=x^{1+y+y^{2}}H_{2}'=x^{3+3(y-1)+(y-1)^{2}}H_{2}'=x^{3}\cdot [x,y]^{3}\cdot [[x,y],y]H_{2}'=s_{c}^{w}s_{2}^{-3}t_{3}^{-1}H_{2}'=t_{3}^{-1}H_{2}'\end{aligned}}} Finally, we combine the results: generally T ~ i ( g G ′ ) = T ~ i ( x G ′ ) j T ~ i ( y G ′ ) ℓ , {\displaystyle {\tilde {T}}_{i}(gG')={\tilde {T}}_{i}(xG')^{j}{\tilde {T}}_{i}(yG')^{\ell },} and in particular, T ~ 1 ( g G ′ ) = s c w j + z ℓ H 1 ′ T ~ 2 ( g G ′ ) = t 3 − j H 2 ′ T ~ i ( g G ′ ) = s 3 2 ℓ H i ′ 3 ≤ i ≤ 4 {\displaystyle {\begin{aligned}{\tilde {T}}_{1}(gG')&=s_{c}^{wj+z\ell }H_{1}'\\\\{\tilde {T}}_{2}(gG')&=t_{3}^{-j}H_{2}'\\\\{\tilde {T}}_{i}(gG')&=s_{3}^{2\ell }H_{i}'&&3\leq i\leq 4\end{aligned}}} To determine the kernels, it remains to solve the equations: s c w j + z ℓ H 1 ′ = H 1 ′ ⇒ { arbitrary j , ℓ and w = z = 0 ℓ = 0 , arbitrary j and w = 0 , z = ± 1 j = 0 , arbitrary ℓ and w = 1 , z = 0 j = ∓ ℓ , w = 1 , z = ± 1 t 3 − j H 2 ′ = H 2 ′ ⇒ j = 0 with arbitrary ℓ s 3 2 ℓ H i ′ = H i ′ ⇒ ℓ = 0 with arbitrary j 3 ≤ i ≤ 4 {\displaystyle {\begin{aligned}s_{c}^{wj+z\ell }H_{1}'=H_{1}'&\Rightarrow {\begin{cases}{\text{arbitrary }}j,\ell {\text{ and }}w=z=0\\\\\ell =0,{\text{arbitrary }}j{\text{ and }}w=0,z=\pm 1\\\j=0,{\text{arbitrary }}\ell {\text{ and }}w=1,z=0\\\j=\mp \ell ,w=1,z=\pm 1\end{cases}}\\\t_{3}^{-j}H_{2}'=H_{2}'&\Rightarrow j=0{\text{ with arbitrary }}\ell \\\s_{3}^{2\ell }H_{i}'=H_{i}'&\Rightarrow \ell =0{\text{ with arbitrary }}j&&3\leq i\leq 4\end{aligned}}} The following equivalences, for any 1 ≤ i ≤ 4 {\displaystyle 1\leq i\leq 4} , finish the justification of the statements: * j , ℓ {\displaystyle j,\ell } both arbitrary ⇔ ker ⁡ ( T i ) = ⟨ x , y , G ′ ⟩ = G ⇔ ϰ ( i ) = 0 {\displaystyle \Leftrightarrow \ker(T_{i})=\langle x,y,G'\rangle =G\Leftrightarrow \varkappa (i)=0} . * j = 0 {\displaystyle j=0} with arbitrary ℓ ⇔ ker ⁡ ( T i ) = ⟨ y , G ′ ⟩ = H 1 ⇔ ϰ ( i ) = 1 {\displaystyle \ell \Leftrightarrow \ker(T_{i})=\langle y,G'\rangle =H_{1}\Leftrightarrow \varkappa (i)=1} , * ℓ = 0 {\displaystyle \ell =0} with arbitrary j ⇔ ker ⁡ ( T i ) = ⟨ x , G ′ ⟩ = H 2 ⇔ ϰ ( i ) = 2 {\displaystyle j\Leftrightarrow \ker(T_{i})=\langle x,G'\rangle =H_{2}\Leftrightarrow \varkappa (i)=2} , * j = ℓ ⇔ ker ⁡ ( T i ) = ⟨ x y , G ′ ⟩ = H 3 ⇔ ϰ ( i ) = 3 {\displaystyle j=\ell \Leftrightarrow \ker(T_{i})=\langle xy,G'\rangle =H_{3}\Leftrightarrow \varkappa (i)=3} , * j = − ℓ ⇔ ker ⁡ ( T i ) = ⟨ x y − 1 , G ′ ⟩ = H 4 ⇔ ϰ ( i ) = 4 {\displaystyle j=-\ell \Leftrightarrow \ker(T_{i})=\langle xy^{-1},G'\rangle =H_{4}\Leftrightarrow \varkappa (i)=4} Consequently, the last three components of the TKT are independent of the parameters w , z , {\displaystyle w,z,} which means that both, the TTT and the TKT, reveal a uni-polarization at the first component. ## Systematic library of SDTs[edit] The aim of this section is to present a collection of structured coclass trees (SCTs) of finite p-groups with parametrized presentations and a succinct summary of invariants. The underlying prime p {\displaystyle p} is restricted to small values p ∈ { 2 , 3 , 5 } {\displaystyle p\in \\{2,3,5\\}} . The trees are arranged according to increasing coclass r ≥ 1 {\displaystyle r\geq 1} and different abelianizations within each coclass. To keep the descendant numbers manageable, the trees are pruned by eliminating vertices of depth bigger than one. Further, we omit trees where stabilization criteria enforce a common TKT of all vertices, since we do not consider such trees as structured any more. The invariants listed include * pre-period and period length, * depth and width of branches, * uni-polarization, TTT and TKT, * σ {\displaystyle \sigma } -groups. We refrain from giving justifications for invariants, since the way how invariants are derived from presentations was demonstrated exemplarily in the section on commutator calculus Figure 5: Structured descendant tree of 2-groups with coclass 1. ### Coclass 1[edit] For each prime p ∈ { 2 , 3 , 5 } {\displaystyle p\in \\{2,3,5\\}} , the unique tree of p-groups of maximal class is endowed with information on TTTs and TKTs, that is, T 1 ( ⟨ 4 , 2 ⟩ ) {\displaystyle {\mathcal {T}}^{1}(\langle 4,2\rangle )} for p = 2 , T 1 ( ⟨ 9 , 2 ⟩ ) {\displaystyle p=2,{\mathcal {T}}^{1}(\langle 9,2\rangle )} for p = 3 {\displaystyle p=3} , and T 1 ( ⟨ 25 , 2 ⟩ ) {\displaystyle {\mathcal {T}}^{1}(\langle 25,2\rangle )} for p = 5 {\displaystyle p=5} . In the last case, the tree is restricted to metabelian 5 {\displaystyle 5} -groups. The 2 {\displaystyle 2} -groups of coclass 1 {\displaystyle 1} in Figure 5 can be defined by the following parametrized polycyclic pc-presentation, quite different from Blackburn's presentation.[10] 2 G c , n ( z , w ) = ⟨ x , y , s 2 , … , s c ∣ x 2 = s c w , y 2 = s c z , s j 2 = s j + 1 s j + 2 for 2 ≤ j ≤ c − 2 , s c − 1 2 = s c , s 2 = [ y , x ] , s j = [ s j − 1 , x ] = [ s j − 1 , y ] for 3 ≤ j ≤ c ⟩ , {\displaystyle {\begin{aligned}G^{c,n}(z,w)=&\langle x,y,s_{2},\ldots ,s_{c}\mid {}\\\&x^{2}=s_{c}^{w},\ y^{2}=s_{c}^{z},\ s_{j}^{2}=s_{j+1}s_{j+2}{\text{ for }}2\leq j\leq c-2,\ s_{c-1}^{2}=s_{c},\\\&s_{2}=[y,x],\ s_{j}=[s_{j-1},x]=[s_{j-1},y]{\text{ for }}3\leq j\leq c\rangle ,\end{aligned}}} where the nilpotency class is c ≥ 3 {\displaystyle c\geq 3} , the order is 2 n {\displaystyle 2^{n}} with n = c + 1 {\displaystyle n=c+1} , and w , z {\displaystyle w,z} are parameters. The branches are strictly periodic with pre-period 1 {\displaystyle 1} and period length 1 {\displaystyle 1} , and have depth 1 {\displaystyle 1} and width 3 {\displaystyle 3} . Polarization occurs for the third component and the TTT is τ = [ ( 1 2 ) , ( 1 2 ) , A ( 2 , c ) ] {\displaystyle \tau =[(1^{2}),(1^{2}),A(2,c)]} , only dependent on c {\displaystyle c} and with cyclic A ( 2 , c ) {\displaystyle A(2,c)} . The TKT depends on the parameters and is ϰ = ( 210 ) {\displaystyle \varkappa =(210)} for the dihedral mainline vertices with w = z = 0 {\displaystyle w=z=0} , ϰ = ( 213 ) {\displaystyle \varkappa =(213)} for the terminal generalized quaternion groups with w = z = 1 {\displaystyle w=z=1} , and ϰ = ( 211 ) {\displaystyle \varkappa =(211)} for the terminal semi dihedral groups with w = 1 , z = 0 {\displaystyle w=1,z=0} . There are two exceptions, the abelian root with τ = [ ( 1 ) , ( 1 ) , ( 1 ) ] {\displaystyle \tau =[(1),(1),(1)]} and ϰ = ( 000 ) {\displaystyle \varkappa =(000)} , and the usual quaternion group with τ = [ ( 2 ) , ( 2 ) , ( 2 ) ] {\displaystyle \tau =[(2),(2),(2)]} and ϰ = ( 123 ) {\displaystyle \varkappa =(123)} . Figure 6: Structured descendant tree of 3-groups with coclass 1. The 3 {\displaystyle 3} -groups of coclass 1 {\displaystyle 1} in Figure 6 can be defined by the following parametrized polycyclic pc-presentation, slightly different from Blackburn's presentation.[10] 3 G a c , n ( z , w ) = ⟨ x , y , s 2 , t 3 , s 3 , … , s c ∣ x 3 = s c w , y 3 = s 3 2 s 4 s c z , t 3 = s c a , s j 3 = s j + 2 2 s j + 3 for 2 ≤ j ≤ c − 3 , s c − 2 3 = s c 2 , s 2 = [ y , x ] , t 3 = [ s 2 , y ] , s j = [ s j − 1 , x ] for 3 ≤ j ≤ c ⟩ , {\displaystyle {\begin{aligned}G_{a}^{c,n}(z,w)=&\langle x,y,s_{2},t_{3},s_{3},\ldots ,s_{c}\mid {}\\\&x^{3}=s_{c}^{w},\ y^{3}=s_{3}^{2}s_{4}s_{c}^{z},\ t_{3}=s_{c}^{a},\ s_{j}^{3}=s_{j+2}^{2}s_{j+3}{\text{ for }}2\leq j\leq c-3,\ s_{c-2}^{3}=s_{c}^{2},\\\&s_{2}=[y,x],\ t_{3}=[s_{2},y],\ s_{j}=[s_{j-1},x]{\text{ for }}3\leq j\leq c\rangle ,\end{aligned}}} where the nilpotency class is c ≥ 5 {\displaystyle c\geq 5} , the order is 3 n {\displaystyle 3^{n}} with n = c + 1 {\displaystyle n=c+1} , and a , w , z {\displaystyle a,w,z} are parameters. The branches are strictly periodic with pre-period 2 {\displaystyle 2} and period length 2 {\displaystyle 2} , and have depth 1 {\displaystyle 1} and width 7 {\displaystyle 7} . Polarization occurs for the first component and the TTT is τ = [ A ( 3 , c − a ) , ( 1 2 ) , ( 1 2 ) , ( 1 2 ) ] {\displaystyle \tau =[A(3,c-a),(1^{2}),(1^{2}),(1^{2})]} , only dependent on c {\displaystyle c} and a {\displaystyle a} . The TKT depends on the parameters and is ϰ = ( 0000 ) {\displaystyle \varkappa =(0000)} for the mainline vertices with a = w = z = 0 , ϰ = ( 1000 ) {\displaystyle a=w=z=0,\varkappa =(1000)} for the terminal vertices with a = 0 , w = 1 , z = 0 , ϰ = ( 2000 ) {\displaystyle a=0,w=1,z=0,\varkappa =(2000)} for the terminal vertices with a = w = 0 , z = ± 1 {\displaystyle a=w=0,z=\pm 1} , and ϰ = ( 0000 ) {\displaystyle \varkappa =(0000)} for the terminal vertices with a = 1 , w ∈ { − 1 , 0 , 1 } , z = 0 {\displaystyle a=1,w\in \\{-1,0,1\\},z=0} . There exist three exceptions, the abelian root with τ = [ ( 1 ) , ( 1 ) , ( 1 ) , ( 1 ) ] {\displaystyle \tau =[(1),(1),(1),(1)]} , the extra special group of exponent 9 {\displaystyle 9} with τ = [ ( 1 2 ) , ( 2 ) , ( 2 ) , ( 2 ) ] {\displaystyle \tau =[(1^{2}),(2),(2),(2)]} and ϰ = ( 1111 ) {\displaystyle \varkappa =(1111)} , and the Sylow 3 {\displaystyle 3} -subgroup of the alternating group A 9 {\displaystyle A_{9}} with τ = [ ( 1 3 ) , ( 1 2 ) , ( 1 2 ) , ( 1 2 ) ] {\displaystyle \tau =[(1^{3}),(1^{2}),(1^{2}),(1^{2})]} . Mainline vertices and vertices on odd branches are σ {\displaystyle \sigma } -groups. Figure 7: Structured descendant tree of metabelian 5-groups with coclass 1. The metabelian 5 {\displaystyle 5} -groups of coclass 1 {\displaystyle 1} in Figure 7 can be defined by the following parametrized polycyclic pc-presentation, slightly different from Miech's presentation.[21] 4 G a c , n ( z , w ) = ⟨ x , y , s 2 , t 3 , s 3 , … , s c ∣ x 5 = s c w , y 5 = s c z , t 3 = s c a , s 2 = [ y , x ] , t 3 = [ s 2 , y ] , s j = [ s j − 1 , x ] for 3 ≤ j ≤ c ⟩ , {\displaystyle {\begin{aligned}G_{a}^{c,n}(z,w)=&\langle x,y,s_{2},t_{3},s_{3},\ldots ,s_{c}\mid {}\\\&x^{5}=s_{c}^{w},\ y^{5}=s_{c}^{z},\ t_{3}=s_{c}^{a},\\\&s_{2}=[y,x],\ t_{3}=[s_{2},y],\ s_{j}=[s_{j-1},x]{\text{ for }}3\leq j\leq c\rangle ,\end{aligned}}} where the nilpotency class is c ≥ 3 {\displaystyle c\geq 3} , the order is 5 n {\displaystyle 5^{n}} with n = c + 1 {\displaystyle n=c+1} , and a , w , z {\displaystyle a,w,z} are parameters. The (metabelian!) branches are strictly periodic with pre-period 3 {\displaystyle 3} and period length 4 {\displaystyle 4} , and have depth 3 {\displaystyle 3} and width 67 {\displaystyle 67} . (The branches of the complete tree, including non-metabelian groups, are only virtually periodic and have bounded width but unbounded depth!) Polarization occurs for the first component and the TTT is τ = [ A ( 5 , c − k ) , ( 1 2 ) 5 ] {\displaystyle \tau =[A(5,c-k),(1^{2})^{5}]} , only dependent on c {\displaystyle c} and the defect of commutativity k {\displaystyle k} . The TKT depends on the parameters and is ϰ = ( 0 6 ) {\displaystyle \varkappa =(0^{6})} for the mainline vertices with a = w = z = 0 , ϰ = ( 10 5 ) {\displaystyle a=w=z=0,\varkappa =(10^{5})} for the terminal vertices with a = 0 , w = 1 , z = 0 , ϰ = ( 20 5 ) {\displaystyle a=0,w=1,z=0,\varkappa =(20^{5})} for the terminal vertices with a = w = 0 , z ≠ 0 {\displaystyle a=w=0,z\neq 0} , and ϰ = ( 0 6 ) {\displaystyle \varkappa =(0^{6})} for the vertices with a ≠ 0 {\displaystyle a\neq 0} . There exist three exceptions, the abelian root with τ = [ ( 1 ) 6 ] {\displaystyle \tau =[(1)^{6}]} , the extra special group of exponent 25 {\displaystyle 25} with τ = [ ( 1 2 ) , ( 2 ) 5 ] {\displaystyle \tau =[(1^{2}),(2)^{5}]} and ϰ = ( 1 6 ) {\displaystyle \varkappa =(1^{6})} , and the group ⟨ 15625 , 631 ⟩ {\displaystyle \langle 15625,631\rangle } with τ = [ ( 1 5 ) , ( 1 2 ) 5 ] {\displaystyle \tau =[(1^{5}),(1^{2})^{5}]} . Mainline vertices and vertices on odd branches are σ {\displaystyle \sigma } -groups. ### Coclass 2[edit] #### Abelianization of type (p,p)[edit] Three coclass trees, T 2 ( ⟨ 243 , 6 ⟩ ) {\displaystyle {\mathcal {T}}^{2}(\langle 243,6\rangle )} , T 2 ( ⟨ 243 , 8 ⟩ ) {\displaystyle {\mathcal {T}}^{2}(\langle 243,8\rangle )} and T 2 ( ⟨ 729 , 40 ⟩ ) {\displaystyle {\mathcal {T}}^{2}(\langle 729,40\rangle )} for p = 3 {\displaystyle p=3} , are endowed with information concerning TTTs and TKTs. Figure 8: First structured descendant tree of 3-groups with coclass 2 and abelianization (3,3). On the tree T 2 ( ⟨ 243 , 6 ⟩ ) {\displaystyle {\mathcal {T}}^{2}(\langle 243,6\rangle )} , the 3 {\displaystyle 3} -groups of coclass 2 {\displaystyle 2} with bicyclic centre in Figure 8 can be defined by the following parametrized polycyclic pc-presentation. [11] 5 G c , n ( z , w ) = ⟨ x , y , s 2 , t 3 , s 3 , … , s c ∣ x 3 = s c w , y 3 = s 3 2 s 4 s c z , s j 3 = s j + 2 2 s j + 3 for 2 ≤ j ≤ c − 3 , s c − 2 3 = s c 2 , t 3 3 = 1 , s 2 = [ y , x ] , t 3 = [ s 2 , y ] , s j = [ s j − 1 , x ] for 3 ≤ j ≤ c ⟩ , {\displaystyle {\begin{aligned}G^{c,n}(z,w)=&\langle x,y,s_{2},t_{3},s_{3},\ldots ,s_{c}\mid {}\\\&x^{3}=s_{c}^{w},\ y^{3}=s_{3}^{2}s_{4}s_{c}^{z},\ s_{j}^{3}=s_{j+2}^{2}s_{j+3}{\text{ for }}2\leq j\leq c-3,\ s_{c-2}^{3}=s_{c}^{2},\ t_{3}^{3}=1,\\\&s_{2}=[y,x],\ t_{3}=[s_{2},y],\ s_{j}=[s_{j-1},x]{\text{ for }}3\leq j\leq c\rangle ,\end{aligned}}} where the nilpotency class is c ≥ 5 {\displaystyle c\geq 5} , the order is 3 n {\displaystyle 3^{n}} with n = c + 2 {\displaystyle n=c+2} , and w , z {\displaystyle w,z} are parameters. The branches are strictly periodic with pre-period 2 {\displaystyle 2} and period length 2 {\displaystyle 2} , and have depth 3 {\displaystyle 3} and width 18 {\displaystyle 18} . Polarization occurs for the first component and the TTT is τ = [ A ( 3 , c ) , ( 1 3 ) , ( 21 ) , ( 21 ) ] {\displaystyle \tau =[A(3,c),(1^{3}),(21),(21)]} , only dependent on c {\displaystyle c} . The TKT depends on the parameters and is ϰ = ( 0122 ) {\displaystyle \varkappa =(0122)} for the mainline vertices with w = z = 0 {\displaystyle w=z=0} , ϰ = ( 2122 ) {\displaystyle \varkappa =(2122)} for the capable vertices with w = 0 , z = ± 1 {\displaystyle w=0,z=\pm 1} , ϰ = ( 1122 ) {\displaystyle \varkappa =(1122)} for the terminal vertices with w = 1 , z = 0 {\displaystyle w=1,z=0} , and ϰ = ( 3122 ) {\displaystyle \varkappa =(3122)} for the terminal vertices with w = 1 , z = ± 1 {\displaystyle w=1,z=\pm 1} . Mainline vertices and vertices on even branches are σ {\displaystyle \sigma } -groups. Figure 9: Second structured descendant tree of 3-groups with coclass 2 and abelianization (3,3). On the tree T 2 ( ⟨ 243 , 8 ⟩ ) {\displaystyle {\mathcal {T}}^{2}(\langle 243,8\rangle )} , the 3 {\displaystyle 3} -groups of coclass 2 {\displaystyle 2} with bicyclic centre in Figure 9 can be defined by the following parametrized polycyclic pc-presentation. [11] 6 G c , n ( z , w ) = ⟨ x , y , t 2 , s 3 , t 3 , … , t c ∣ y 3 = s 3 t c w , x 3 = t 3 t 4 2 t 5 t c z , t j 3 = t j + 2 2 t j + 3 for 2 ≤ j ≤ c − 3 , t c − 2 3 = t c 2 , s 3 3 = 1 , t 2 = [ y , x ] , s 3 = [ t 2 , x ] , t j = [ t j − 1 , y ] for 3 ≤ j ≤ c ⟩ , {\displaystyle {\begin{aligned}G^{c,n}(z,w)=&\langle x,y,t_{2},s_{3},t_{3},\ldots ,t_{c}\mid {}\\\&y^{3}=s_{3}t_{c}^{w},\ x^{3}=t_{3}t_{4}^{2}t_{5}t_{c}^{z},\ t_{j}^{3}=t_{j+2}^{2}t_{j+3}{\text{ for }}2\leq j\leq c-3,\ t_{c-2}^{3}=t_{c}^{2},\ s_{3}^{3}=1,\\\&t_{2}=[y,x],\ s_{3}=[t_{2},x],\ t_{j}=[t_{j-1},y]{\text{ for }}3\leq j\leq c\rangle ,\end{aligned}}} where the nilpotency class is c ≥ 6 {\displaystyle c\geq 6} , the order is 3 n {\displaystyle 3^{n}} with n = c + 2 {\displaystyle n=c+2} , and w , z {\displaystyle w,z} are parameters. The branches are strictly periodic with pre-period 2 {\displaystyle 2} and period length 2 {\displaystyle 2} , and have depth 3 {\displaystyle 3} and width 16 {\displaystyle 16} . Polarization occurs for the second component and the TTT is τ = [ ( 21 ) , A ( 3 , c ) , ( 21 ) , ( 21 ) ] {\displaystyle \tau =[(21),A(3,c),(21),(21)]} , only dependent on c {\displaystyle c} . The TKT depends on the parameters and is ϰ = ( 2034 ) {\displaystyle \varkappa =(2034)} for the mainline vertices with w = z = 0 {\displaystyle w=z=0} , ϰ = ( 2134 ) {\displaystyle \varkappa =(2134)} for the capable vertices with w = 0 , z = ± 1 {\displaystyle w=0,z=\pm 1} , ϰ = ( 2234 ) {\displaystyle \varkappa =(2234)} for the terminal vertices with w = 1 , z = 0 {\displaystyle w=1,z=0} , and ϰ = ( 2334 ) {\displaystyle \varkappa =(2334)} for the terminal vertices with w = 1 , z = ± 1 {\displaystyle w=1,z=\pm 1} . Mainline vertices and vertices on even branches are σ {\displaystyle \sigma } -groups. #### Abelianization of type (p2,p)[edit] T 2 ( ⟨ 16 , 3 ⟩ ) {\displaystyle {\mathcal {T}}^{2}(\langle 16,3\rangle )} and T 2 ( ⟨ 16 , 4 ⟩ ) {\displaystyle {\mathcal {T}}^{2}(\langle 16,4\rangle )} for p = 2 {\displaystyle p=2} , T 2 ( ⟨ 243 , 15 ⟩ ) {\displaystyle {\mathcal {T}}^{2}(\langle 243,15\rangle )} and T 2 ( ⟨ 243 , 17 ⟩ ) {\displaystyle {\mathcal {T}}^{2}(\langle 243,17\rangle )} for p = 3 {\displaystyle p=3} . #### Abelianization of type (p,p,p)[edit] T 2 ( ⟨ 16 , 11 ⟩ ) {\displaystyle {\mathcal {T}}^{2}(\langle 16,11\rangle )} for p = 2 {\displaystyle p=2} , and T 2 ( ⟨ 81 , 12 ⟩ ) {\displaystyle {\mathcal {T}}^{2}(\langle 81,12\rangle )} for p = 3 {\displaystyle p=3} . ### Coclass 3[edit] #### Abelianization of type (p2,p)[edit] T 3 ( ⟨ 729 , 13 ⟩ ) {\displaystyle {\mathcal {T}}^{3}(\langle 729,13\rangle )} , T 3 ( ⟨ 729 , 18 ⟩ ) {\displaystyle {\mathcal {T}}^{3}(\langle 729,18\rangle )} and T 3 ( ⟨ 729 , 21 ⟩ ) {\displaystyle {\mathcal {T}}^{3}(\langle 729,21\rangle )} for p = 3 {\displaystyle p=3} . #### Abelianization of type (p,p,p)[edit] T 3 ( ⟨ 32 , 35 ⟩ ) {\displaystyle {\mathcal {T}}^{3}(\langle 32,35\rangle )} and T 3 ( ⟨ 64 , 181 ⟩ ) {\displaystyle {\mathcal {T}}^{3}(\langle 64,181\rangle )} for p = 2 {\displaystyle p=2} , T 3 ( ⟨ 243 , 38 ⟩ ) {\displaystyle {\mathcal {T}}^{3}(\langle 243,38\rangle )} and T 3 ( ⟨ 243 , 41 ⟩ ) {\displaystyle {\mathcal {T}}^{3}(\langle 243,41\rangle )} for p = 3 {\displaystyle p=3} . Figure 10: Minimal discriminants for the first ASCT of 3-groups with coclass 2 and abelianization (3,3). ## Arithmetical applications[edit] In algebraic number theory and class field theory, structured descendant trees (SDTs) of finite p-groups provide an excellent tool for * visualizing the location of various non-abelian p-groups G ( K ) {\displaystyle G(K)} associated with algebraic number fields K {\displaystyle K} , * displaying additional information about the groups G ( K ) {\displaystyle G(K)} in labels attached to corresponding vertices, and * emphasizing the periodicity of occurrences of the groups G ( K ) {\displaystyle G(K)} on branches of coclass trees. For instance, let p {\displaystyle p} be a prime number, and assume that F p 2 ( K ) {\displaystyle F_{p}^{2}(K)} denotes the second Hilbert p-class field of an algebraic number field K {\displaystyle K} , that is the maximal metabelian unramified extension of K {\displaystyle K} of degree a power of p {\displaystyle p} . Then the second p-class group G p 2 ( K ) = G a l ( F p 2 ( K ) | K ) {\displaystyle G_{p}^{2}(K)=\mathrm {Gal} (F_{p}^{2}(K)|K)} of K {\displaystyle K} is usually a non-abelian p-group of derived length 2 {\displaystyle 2} and frequently permits to draw conclusions about the entire p-class field tower of K {\displaystyle K} , that is the Galois group G p ∞ ( K ) = G a l ( F p ∞ ( K ) | K ) {\displaystyle G_{p}^{\infty }(K)=\mathrm {Gal} (F_{p}^{\infty }(K)|K)} of the maximal unramified pro-p extension F p ∞ ( K ) {\displaystyle F_{p}^{\infty }(K)} of K {\displaystyle K} . Given a sequence of algebraic number fields K {\displaystyle K} with fixed signature ( r 1 , r 2 ) {\displaystyle (r_{1},r_{2})} , ordered by the absolute values of their discriminants d = d ( K ) {\displaystyle d=d(K)} , a suitable structured coclass tree (SCT) T {\displaystyle {\mathcal {T}}} , or also the finite sporadic part G 0 ( p , r ) {\displaystyle {\mathcal {G}}_{0}(p,r)} of a coclass graph G ( p , r ) {\displaystyle {\mathcal {G}}(p,r)} , whose vertices are entirely or partially realized by second p-class groups G p 2 ( K ) {\displaystyle G_{p}^{2}(K)} of the fields K {\displaystyle K} is endowed with additional arithmetical structure when each realized vertex V ∈ T {\displaystyle V\in {\mathcal {T}}} , resp. V ∈ G 0 ( p , r ) {\displaystyle V\in {\mathcal {G}}_{0}(p,r)} , is mapped to data concerning the fields K {\displaystyle K} such that V = G p 2 ( K ) {\displaystyle V=G_{p}^{2}(K)} . Figure 11: Minimal discriminants for the second ASCT of 3-groups with coclass 2 and abelianization (3,3). ### Example[edit] To be specific, let p = 3 {\displaystyle p=3} and consider complex quadratic fields K ( d ) = Q ( d ) {\displaystyle K(d)=\mathbb {Q} ({\sqrt {d}})} with fixed signature ( 0 , 1 ) {\displaystyle (0,1)} having 3 {\displaystyle 3} -class groups with type invariants ( 3 , 3 ) {\displaystyle (3,3)} . See OEIS A242863 [1]. Their second 3 {\displaystyle 3} -class groups G 3 2 ( K ) {\displaystyle G_{3}^{2}(K)} have been determined by D. C. Mayer [17] for the range − 10 6 < d < 0 {\displaystyle -10^{6} 3 {\displaystyle p>3} , the stem of Φ 6 {\displaystyle \Phi _{6}} consists of p + 7 {\displaystyle p+7} regular p-groups and reveals a rather uniform behaviour with respect to TKTs and TTTs, but the seven 3 {\displaystyle 3} -groups in the stem of Φ 6 {\displaystyle \Phi _{6}} are irregular. We emphasize that there also exist several ( 3 {\displaystyle 3} for p = 3 {\displaystyle p=3} and 4 {\displaystyle 4} for p > 3 {\displaystyle p>3} ) infinitely capable vertices in the stem of Φ 6 {\displaystyle \Phi _{6}} which are partially roots of coclass trees. However, here we focus on the sporadic vertices which are either isolated Schur σ {\displaystyle \sigma } -groups ( 2 {\displaystyle 2} for p = 3 {\displaystyle p=3} and p + 1 {\displaystyle p+1} for p > 3 {\displaystyle p>3} ) or roots of finite trees within G 0 ( p , 2 ) {\displaystyle {\mathcal {G}}_{0}(p,2)} ( 2 {\displaystyle 2} for each p ≥ 3 {\displaystyle p\geq 3} ). For p > 3 {\displaystyle p>3} , the TKT of Schur σ {\displaystyle \sigma } -groups is a permutation whose cycle decomposition does not contain transpositions, whereas the TKT of roots of finite trees is a compositum of disjoint transpositions having an even number ( 0 {\displaystyle 0} or 2 {\displaystyle 2} ) of fixed points. We endow the forest G 0 ( p , 2 ) {\displaystyle {\mathcal {G}}_{0}(p,2)} (a finite union of descendant trees) with additional arithmetical structure by attaching the minimal absolute discriminant min { | d | ∣ V = G p 2 ( K ( d ) ) } {\displaystyle \min\\{|d|\mid V=G_{p}^{2}(K(d))\\}} to each realized vertex V ∈ G 0 ( p , 2 ) {\displaystyle V\in {\mathcal {G}}_{0}(p,2)} . The resulting structured sporadic coclass graph is shown in Figure 13 for p = 3 {\displaystyle p=3} , in Figure 14 for p = 5 {\displaystyle p=5} , and in Figure 15 for p = 7 {\displaystyle p=7} . ## References[edit] 1. ^ a b c d e f g h i Huppert, B. (1979). Endliche Gruppen I. Grundlehren der mathematischen Wissenschaften, Vol. 134, Springer-Verlag Berlin Heidelberg New York. 2. ^ a b c d e Schur, I. (1902). "Neuer Beweis eines Satzes über endliche Gruppen". Sitzungsb. Preuss. Akad. Wiss.: 1013–1019. 3. ^ a b c d Artin, E. (1929). "Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetz". Abh. Math. Sem. 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