Socle

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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

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 nontrivial semidirect product of Z4 and Z4 Nontrivial semidirect product of Z4 and Z4 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
A3 in S3 Symmetric group:S3 Cyclic group:Z3 Cyclic group:Z2
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