# Let $G$ be a group s.t. $|G|=pqr$ where $p,q,r$ are primes that need not be distinct. Prove that $G$ is soluble.

Let $$G$$ be a group s.t. $$|G|=pqr$$ where $$p,q,r$$ are primes that need not be distinct. Prove that $$G$$ is soluble.

So, I don't know whether I should handle this case by case and try to get the Sylow theorems involved or if there is an easier way to do this, a slick trick I am not seeing perhaps. Does anyone have any insights?

• Do you know that groups of order $pqr$ are not simple? – Mark Apr 22 at 21:30
• no i did not know that – Mathematical Mushroom Apr 22 at 21:36