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Equivalence of definitions of Fitting-free group

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

Facts used

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