Let $G$ be a group s.t. $|G|=pqr$, where $p,q,r$ are primes that are not necessarily distinct. Show that $G$ is soluable.
Unless there is a more general method to do this, I was going to break it up into cases when all thee primes are different, when two are the same, and when they are all distinct.
For a group to be soluable, then either it's derivied series has to terminate in the trivial subgroup, or there has to be a series such that every factor is abelian. I've been trying to use both definitions for the case when $G$ is three distinct primes but I am stuck an need some new insights.
