A non-trivial normal subgroup $H$ such that between it and the identity subgroup there are no other normal subgroups of the group. Not all groups have a minimal normal subgroup. If the group is finite, then any minimal normal subgroup of it is a direct product of isomorphic simple groups. If a minimal normal subgroup exists and is unique, then it is called a monolith (sometimes, a socle), and the group itself is called a monolithic group.

#### References[edit]

[1] |  A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1960) (Translated from Russian)

|

#### Comments[edit]

I.e. a minimal normal subgroup is a non-trivial normal subgroup that is minimal in the set of all such subgroups, ordered by inclusion.

#### References[edit]

[a1] |  D.J.S. Robinson, "A course in the theory of groups" , Springer (1980)

|