# Fitting subgroup is normal-isomorph-free in finite

From Groupprops

This article gives the statement, and possibly proof, of the fact that in any finite group, the subgroup obtained by applying a given subgroup-defining function (i.e., Fitting subgroup) always satisfies a particular subgroup property (i.e., normal-isomorph-free subgroup)

View all such subgroup property satisfactions OR View more information on subgroup-defining functions in finite groups

## Contents

## Statement

In a finite group, the Fitting subgroup is **normal-isomorph-free**: there is no other normal subgroup of the whole group isomorphic to it.

## Related facts

- Perfect core is homomorph-containing
- Solvable core is normal-isomorph-free in finite
- Fitting subgroup not is isomorph-free in finite

## Proof

### Proof idea

In a finite group, the Fitting subgroup is the unique largest nilpotent normal subgroup. Any normal subgroup isomorphic to it would also be a nilpotent normal subgroup, and hence, in particular, would be contained in it. Since the group is finite, this forces that any normal subgroup isomorphic to the Fitting subgroup must be equal to it.