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From MathWorld on solvable group:

A solvable group is a group having a normal series such that each normal factor is Abelian. The special case of a solvable finite group is a group whose composition indices are all prime numbers.

But why is that a

*special*case? The way I understand it: the normal series can always be made such that all composition factors are simple, but then the composition factors are both simple and Abelian, and hence (isomorphic to) [itex]\mathbb Z_p[/itex], i.e. the composition index is p (= prime)...