# Socle

From Groupprops

This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup

View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions

## Definition

The **socle** of a group can be defined as the subgroup generated by all minimal normal subgroups, i.e., the join of all minimal normal subgroups.

## Formalisms

### In terms of the join-all operator

This property is obtained by applying the join-all operator to the property: minimal normal subgroup

View other properties obtained by applying the join-all operator

## Facts

- Socle of finite group is internal direct product of simple subnormal subgroups: Since minimal normal implies characteristically simple, the socle is generated by characteristically simple groups. In a finite group, a characteristically simple group is a direct product of simple groups, so the socle is generated by simple groups. It can further be shown that the socle is an internal direct product of a collection of simple subnormal subgroups.
- Socle equals Omega-1 of center for nilpotent p-group: If is a group of prime power order, or more generally in a (possibly infinite) nilpotent p-group, the socle equals , i.e., the set of elements of prime order in the center of (along with the identity element). This follows from the fact that minimal normal implies central in nilpotent.
- If is a finite solvable group, then the socle is a product of elementary abelian -groups for a collection of primes dividing the order of (though this may not include
*all*primes dividing the order of ). This follows from the fact that minimal normal implies elementary abelian in finite solvable. (When is nilpotent,*all*primes dividing its order are included).

## Examples

### Groups of prime power order

Here, the socle is Omega-1 of the center:

### Finite solvable groups that are not nilpotent

Here, the socle is a product of elementary abelian groups for some of the primes dividing the order of the group:

### Groups that are not solvable

Here, the socle is a product of simple groups, but we cannot say a priori whether it will comprise only simple abelian groups, simple non-abelian groups, or both.

Group part | Subgroup part | Quotient part | |
---|---|---|---|

A5 in S5 | Symmetric group:S5 | Alternating group:A5 | Cyclic group:Z2 |

Center of special linear group:SL(2,5) | Special linear group:SL(2,5) | Cyclic group:Z2 | Alternating group:A5 |

## Subgroup properties

### Properties satisfied

### Properties not satisfied

These are properties that are not *always* satisfied by the socle. They may be satisfied by the socle in a particular group.

Property | Meaning | Proof of dissatisfaction |
---|---|---|

fully invariant subgroup | invariant under all endomorphisms | socle not is fully invariant |

transitively normal subgroup | every normal subgroup of the subgroup is normal in the whole group | socle not is transitively normal |

## Computation

### GAP command

The command for computing this subgroup-defining function in:Groups, Algorithms and Programming(GAP) isSocle

View other GAP-computable subgroup-defining functions