# Solvable group: decomposable in prime order groups?

Hey!

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) $\mathbb Z_p$, i.e. the composition index is p (= prime)...