# Socle

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 $G$ is a group of prime power order, or more generally in a (possibly infinite) nilpotent p-group, the socle equals $\Omega_1(Z(G))$, i.e., the set of elements of prime order in the center of $G$ (along with the identity element). This follows from the fact that minimal normal implies central in nilpotent.
• If $G$ is a finite solvable group, then the socle is a product of elementary abelian $p$-groups for a collection of primes dividing the order of $G$ (though this may not include all primes dividing the order of $G$). This follows from the fact that minimal normal implies elementary abelian in finite solvable. (When $G$ is nilpotent, all primes dividing its order are included).

## Examples

### Groups of prime power order

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

Group partSubgroup partQuotient part
Center of dihedral group:D16Dihedral group:D16Cyclic group:Z2Dihedral group:D8
Center of dihedral group:D8Dihedral group:D8Cyclic group:Z2Klein four-group
Center of direct product of D8 and Z2Direct product of D8 and Z2Klein four-groupKlein four-group
Center of quaternion groupQuaternion groupCyclic group:Z2Klein four-group
Center of semidihedral group:SD16Semidihedral group:SD16Cyclic group:Z2Dihedral group:D8
Derived subgroup of M16M16Cyclic group:Z2Direct product of Z4 and Z2

### 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:

Group partSubgroup partQuotient part
Center of binary octahedral groupBinary octahedral groupCyclic group:Z2Symmetric group:S4
Center of general linear group:GL(2,3)General linear group:GL(2,3)Cyclic group:Z2Symmetric group:S4
Center of special linear group:SL(2,3)Special linear group:SL(2,3)Cyclic group:Z2Alternating group:A4
Klein four-subgroup of alternating group:A4Alternating group:A4Klein four-groupCyclic group:Z3

### 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 partSubgroup partQuotient part
A5 in S5Symmetric group:S5Alternating group:A5Cyclic group:Z2
Center of special linear group:SL(2,5)Special linear group:SL(2,5)Cyclic group:Z2Alternating group:A5

## Subgroup properties

### Properties satisfied

Property Meaning Proof of satisfaction
normal subgroup invariant under all inner automorphisms socle is normal
characteristic subgroup invariant under all automorphisms socle is characteristic
strictly characteristic subgroup invariant under all surjective endomorphisms socle is strictly characteristic
normality-preserving endomorphism-invariant subgroup invariant under all normality-preserving endomorphisms socle is normality-preserving endomorphism-invariant
weakly normal-homomorph-containing subgroup contains any image under a homomorphism from the subgroup to the whole group that sends normal subgroups of the whole group contained in the subgroup to normal subgroups of the whole group socle is weakly normal-homomorph-containing
direct projection-invariant subgroup invariant under all projections to direct factors socle is direct projection-invariant
finite direct power-closed characteristic subgroup in any finite direct power of the whole group, the corresponding power of the socle is characteristic socle is finite direct power-closed characteristic

### 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) is:Socle
View other GAP-computable subgroup-defining functions