Let $G$ be a group and $|G|=pqr$, where $p,q,r$ are prime numbers that are not necessarily distinct. Show that $G$ is solvable. I try to discuss the classification of groups of order $pqr$ and I also know a group of order $pq$ is solvable but I can't prove it.
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If they are all equal, it’s a $p$-group, If they are all different, find a normal Sylow subgroup. If $p\lt q=r$, find a normal Sylow subgroup. The only potentialy troublesome case is $p=q\lt r\lt p^2$. – Arturo Magidin Aug 11 '20 at 14:22
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1(The cases where they are not distinct are much simpler. Here is $p^2q$: https://math.stackexchange.com/questions/765267/groups-with-g-p2q-prove-that-if-p-and-q-are-primes-then-there-are-n) – David A. Craven Aug 11 '20 at 14:28