Socle is strictly characteristic

From Groupprops
Jump to: navigation, search
This article gives the statement, and possibly proof, of the fact that for any group, the subgroup obtained by applying a given subgroup-defining function (i.e., socle) always satisfies a particular subgroup property (i.e., strictly characteristic subgroup)}
View subgroup property satisfactions for subgroup-defining functions | View subgroup property dissatisfactions for subgroup-defining functions


The socle of a group (defined as the subgroup generated by all minimal normal subgroups) is a strictly characteristic subgroup: any surjective endomorphism of the whole group sends the socle to within itself.

Related facts

Facts used

  1. Socle is normality-preserving endomorphism-invariant
  2. Normality-preserving endomorphism-invariant implies strictly characteristic


The proof follows directly from facts (1) and (2).