group socle

The socle of a group is the subgroupMathworldPlanetmathPlanetmath generated by all minimal normal subgroups. Because the product of normal subgroupsMathworldPlanetmath is a subgroup, it follows we can remove the word “generated” and replace it by “product.” So the socle of a group is now the product of its minimal normal subgroups. This description can be further refined with a few observations.

Proposition 1.

If M and N are minimal normal subgroups then M and N centralize each other.


Given two distinct minimal normal subgroup M and N, [M,N] is contained in N and M as both are normal. Thus [M,N]MN. But M and N are distinct minimal normal subgroups and MN is normal so MN=1 thus [M,N]=1. ∎

Proposition 2.

The socle of a finite groupMathworldPlanetmath is a direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of minimal normal subgroups.


Let S be the socle of G. We already know S is the product of its minimal normal subgroups, so let us assume S=N1Nk where each Ni is a distinct minimal normal subgroup of G. Thus N1N2=1 and N1N2 clearly contains N1 and N2. Now suppose we extend this to a subsquence Ni1=N1,Ni2=N2,Ni3,,Nij where


for 1k<j and NiNi1Nij for all 1iij. Then consider Nij+1.

As Nij+1 is a minimal normal subgroup and Ni1Nij is a normal subgroup, Nij+1 is either contained in Ni1Nij or intersects trivially. If Nij+1 is contained in Ni1Nij then skip to the next Ni, otherwise set it to be Nij+1. The result is a squence Ni1,,Nij of minimal normal subgroups where S=Ni1Nis and


As we have already seen distinct minimal normal subgroups centralize each other we conclude that S=Ni1××Nis. ∎

Proposition 3.

A minimal normal subgroup is characteristically simple, so if it is finite then it is a product of isomorphicPlanetmathPlanetmathPlanetmath simple groupsMathworldPlanetmathPlanetmath.


If M is a minimal normal subgroup of G and 1<C<M is characteristicPlanetmathPlanetmath in M, then C is normal in G which contradicts the minimality of M. Thus M is characteristically simple. ∎

Corollary 4.

The socle of a finite group is a direct product of simple groups.


As each Nij is characteristically simple each Nij is a direct product of isomorphic simple groups, thus S is a direct product simple groups. ∎

Title group socle
Canonical name GroupSocle
Date of creation 2013-03-22 15:55:12
Last modified on 2013-03-22 15:55:12
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 14
Author Algeboy (12884)
Entry type Definition
Classification msc 20E34
Synonym socle
Defines socle