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 ${\mathrm{\Omega }}_1(Z(G))$, 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:D8	Dihedral group:D8	Cyclic group:Z2	Klein four-group

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.

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