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

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