Socle

This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
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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 part	Subgroup part	Quotient part
Center of dihedral group:D16	Dihedral group:D16	Cyclic group:Z2	Dihedral group:D8
Center of dihedral group:D8	Dihedral group:D8	Cyclic group:Z2	Klein four-group
Center of direct product of D8 and Z2	Direct product of D8 and Z2	Klein four-group	Klein four-group
Center of quaternion group	Quaternion group	Cyclic group:Z2	Klein four-group
Center of semidihedral group:SD16	Semidihedral group:SD16	Cyclic group:Z2	Dihedral group:D8
Derived subgroup of M16	M16	Cyclic group:Z2	Direct 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 part	Subgroup part	Quotient part
Center of binary octahedral group	Binary octahedral group	Cyclic group:Z2	Symmetric group:S4
Center of general linear group:GL(2,3)	General linear group:GL(2,3)	Cyclic group:Z2	Symmetric group:S4
Center of special linear group:SL(2,3)	Special linear group:SL(2,3)	Cyclic group:Z2	Alternating group:A4
Klein four-subgroup of alternating group:A4	Alternating group:A4	Klein four-group	Cyclic 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 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

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