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.
| [1] | A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1960) (Translated from Russian) |
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.
| [a1] | D.J.S. Robinson, "A course in the theory of groups" , Springer (1980) |