Socle is strictly characteristic

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

Statement

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

• Monolith is strictly characteristic: This is a special case where there is a unique minimal normal subgroup, and this is contained in every nontrivial normal subgroup.
• Baer norm is strictly characteristic
• Fitting subgroup is strictly characteristic
• Solvable core is strictly characteristic
• Center is strictly characteristic

Facts used

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

Proof

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