# Solvable group: decomposable in prime order groups?

1. Sep 17, 2013

### nonequilibrium

Hey!

From MathWorld on solvable group:
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)...

2. Sep 17, 2013

### 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 $\mathbb{Z}$).

3. Sep 18, 2013

### nonequilibrium

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