# 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 $[M,N]\le M\cap N$. 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 a subsquence ${N}_{{i}_{1}}={N}_{1},{N}_{{i}_{2}}={N}_{2},{N}_{{i}_{3}},\mathrm{\dots},{N}_{{i}_{j}}$ 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 a normal 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 subgroups where $S={N}_{{i}_{1}}\mathrm{\cdots}{N}_{{i}_{s}}$ 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 other we 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. ∎

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 |