# Socle is strictly characteristic

From Groupprops

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

## Contents

## 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

- Socle is normality-preserving endomorphism-invariant
- Normality-preserving endomorphism-invariant implies strictly characteristic

## Proof

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