# 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 $N$ centralize each other.

###### Proof.

Given two distinct minimal normal subgroup $M$ and $N$, $[M,N]$ is contained in $N$ and $M$ as both are normal. Thus $[M,N]\leq M\cap N$. 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 $S=N_{1}\cdots N_{k}$ 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 a subsquence $N_{i_{1}}=N_{1},N_{i_{2}}=N_{2},N_{i_{3}},\dots,N_{i_{j}}$ where

 $N_{i_{k}}\cap(N_{i_{1}}\cdots N_{i_{k-1}})=1$

for $1\leq k and $N_{i}\leq N_{i_{1}}\cdots N_{i_{j}}$ for all $1\leq i\leq i_{j}$. Then consider $N_{i_{j}+1}$.

As $N_{i_{j}+1}$ is a minimal normal subgroup and $N_{i_{1}}\cdots N_{i_{j}}$ is a normal subgroup, $N_{i_{j}+1}$ is either contained in $N_{i_{1}}\cdots N_{i_{j}}$ or intersects trivially. If $N_{i_{j}+1}$ is contained in $N_{i_{1}}\cdots N_{i_{j}}$ then skip to the next $N_{i}$, otherwise set it to be $N_{i_{j+1}}$. The result is a squence $N_{i_{1}},\dots,N_{i_{j}}$ of minimal normal subgroups where $S=N_{i_{1}}\cdots N_{i_{s}}$ and

 $N_{i_{j}}\cap(N_{i_{1}}\cdots N_{i_{j-1}})=1,\quad 1\leq j\leq s.$

As we have already seen distinct minimal normal subgroups centralize each other we conclude that $S=N_{i_{1}}\times\cdots\times N_{i_{s}}$. ∎

###### 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 $1 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. ∎

Title group socle GroupSocle 2013-03-22 15:55:12 2013-03-22 15:55:12 Algeboy (12884) Algeboy (12884) 14 Algeboy (12884) Definition msc 20E34 socle socle