# Fitting subgroup not is isomorph-free

From Groupprops

This article gives the statement, and possibly proof, of the fact that for a group, the subgroup obtained by applying a given subgroup-defining function (i.e., Fitting subgroup) doesnotalways satisfy a particular subgroup property (i.e., isomorph-free 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 need not be an isomorph-free subgroup.

## Related facts

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

## Proof

### Example of the symmetric group

`Further information: symmetric group:S4`

Let be the symmetric group on the set . The Fitting subgroup of this is a Klein-four group described as:

.

is not isomorph-free in : it is isomorphic to the group:

.

### A more generic example: the von Dyck group

**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE]

### Another generic example: the product of a Fitting-free group with a nilpotent group

Suppose is a nilpotent group and is a Fitting-free group, containing a subgroup isomorphic to . Let . The Fitting subgroup of is . However, this subgroup is isomorphic to .

An example might be to take as any non-Abelian simple group, and as isomorphic to an Abelian subgroup of . For instance, is the alternating group on five letters and is a cyclic group of order two.