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

Symbol-free definition

The socle of a group is defined as the subgroup generated by all minimal normal subgroups.

Definition with symbols

(fillin)

Group properties satisfied

The socle of a group is a direct product of simple groups. Further, any group that is the direct product of simple groups is its own socle.

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

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