# Fitting subgroup 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., Fitting subgroup) 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 Fitting subgroup of a group (i.e., the subgroup generated by all nilpotent normal subgroups) is a strictly characteristic subgroup: any surjective endomorphism of the whole group sends it to within itself.

## Related facts

Similar results include:

## Facts used

- Fitting subgroup is weakly normal-homomorph-containing
- Weakly normal-homomorph-containing implies strictly characteristic

## Proof

The proof follows directly by piecing together facts (1) and (2).