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Minimal normal subgroup

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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

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


Comments

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

[a1] D.J.S. Robinson, "A course in the theory of groups" , Springer (1980)
How to Cite This Entry:
Minimal normal subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Minimal_normal_subgroup&oldid=32018
This article was adapted from an original article by A.L. Shmel'kin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article