Solvable group: decomposable in prime order groups?

[nonequilibrium]

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)...

 

[fzero]

It is only for a finite group that you are guaranteed to have a composition series. For an infinite group, there may be no normal series where the subgroups are maximal. For instance, $\mathbb{Z}$ cannot have a composition series, since it is not itself simple (every subgroup of $\mathbb{Z}$ is itself isomorphic to $\mathbb{Z}$).

 

[nonequilibrium]

Thank you! I see, so for a finite group the "special case" is always true; that clarifies!