Prove G is a solvable group

Hi there,

I am trying to solve this problem:

Suppose G is a finite group, and the order of G is , where are primes such that . Show that G is a solvable group.

I've tried various approaches through Sylow groups and normal subgroups etc but haven't been able to do it, can anyone lend a hand?

Many thanks!

Let's look at the number of Sylow -subgroups. By the Sylow theorems, since and , must be 1. Thus, the Sylow -subgroup, is normal in . Now consider

.

Any group of order (for p prime) is abelian, and since is abelian.