# Equivalence of definitions of Fitting-free group

This article gives a proof/explanation of the equivalence of multiple definitions for the term Fitting-free group

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This fact is an application of the following pivotal fact/result/idea:characteristic of normal implies normal

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This article defines a replacement theorem

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## Contents

## The definitions that we have to prove as equivalent

The three definitions are:

- There is no nontrivial Abelian normal subgroup
- There is no nontrivial nilpotent normal subgroup
- There is no nontrivial solvable normal subgroup

## Definitions used

### Abelian group

`Further information: Abelian group`

### Nilpotent group

`Further information: Nilpotent group`

### Solvable group

`Further information: Solvable group`

## Facts used

- any Abelian group is nilpotent and any nilpotent group is solvable
- Any characteristic subgroup of a normal subgroup is normal
- Any solvable group contains a nontrivial characteristic Abelian subgroup: the penultimate term of its derived series

## Proof

Clearly, (3) implies (2) implies (1), so we need to show that (1) implies (3). In other words, we need to show that *if* there exists a nontrivial solvable normal subgroup, then there exists a nontrivial Abelian normal subgroup.

The idea is as follows:

- Start with a nontrivial solvable normal subgroup
- Take the penultimate (second last) term of its derived series. This is a nontrivial Abelian characteristic subgroup of the solvable normal subgroup
- Use the fact that a characteristic subgroup of a normal subgroup is normal