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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The definitions that we have to prove as equivalent

The three definitions are:

1. There is no nontrivial Abelian normal subgroup
2. There is no nontrivial nilpotent normal subgroup
3. 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

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