Difference between revisions of "Minimal normal subgroup"
From Encyclopedia of Mathematics
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| − | A non-trivial [[Normal subgroup|normal subgroup]] | + | {{TEX|done}} |
| + | A non-trivial [[Normal subgroup|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==== | ====References==== | ||
Latest revision as of 06:11, 1 May 2014
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
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