group socle

The socle of a group is the subgroup generated by all minimal normal subgroups.Because the product of normal subgroups 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 $N$ centralizeeach other.

Proof.

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

Proposition 2.

The socle of a finite group is a direct product of minimal normal subgroups.

Proof.

Let $S$ be the socle of $G$. We already know $S$ is the product of its minimal normal subgroups, so let us assume $S=N_1\mathrm{\cdots }{N}_k$ where each $N_i$ is a distinct minimal normal subgroup of $G$. Thus $N_1\cap N_2=1$ and$N_1{N}_2$ clearly contains $N_1$ and $N_2$. Now suppose we extend this to asubsquence $N_{i_1}=N_1,N_{i_2}=N_2,N_{i_3},\mathrm{\dots },N_{i_j}$ where

$$N_{i_k}\cap (N_{i_1}\mathrm{\cdots }{N}_{i_{k-1}})=1$$

for and $N_i\le N_{i_1}\mathrm{\cdots }{N}_{i_j}$ for all $1\le i\le i_j$.Then consider $N_{i_j+1}$.

As $N_{i_j+1}$ is a minimal normal subgroup and $N_{i_1}\mathrm{\cdots }{N}_{i_j}$ is anormal subgroup, $N_{i_j+1}$ is either contained in $N_{i_1}\mathrm{\cdots }{N}_{i_j}$or intersects trivially. If $N_{i_j+1}$ is contained in $N_{i_1}\mathrm{\cdots }{N}_{i_j}$ then skip to the next $N_i$, otherwise set it to be $N_{i_{j+1}}$.The result is a squence $N_{i_1},\mathrm{\dots },N_{i_j}$ of minimal normal subgroupswhere $S=N_{i_1}\mathrm{\cdots }{N}_{i_s}$ and

$$N_{i_j}\cap (N_{i_1}\mathrm{\cdots }{N}_{i_{j-1}})=1,1\le j\le s.$$

As we have already seen distinct minimal normal subgroups centralize each otherwe conclude that $S=N_{i_1}\times \mathrm{\cdots }\times N_{i_s}$.∎

Proposition 3.

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

Proof.

If $M$ is a minimal normal subgroup of $G$ and is characteristic 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.

Proof.

As each $N_{i_j}$ is characteristically simple each $N_{i_j}$ is a direct product of isomorphic simple groups, thus $S$ is a direct product simple groups.∎