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I know for a fact that if a group $G$ has order $pqr$ with $p,q,r$ distinct primes, then $G$ is solvable.

Most proofs I see of this are very ugly, and require a lot of case checking to show most cases lead to contradictions.

So is there a relatively "nice" proof of this fact, or am I too optimistic is asking for such thing?

2 Answers 2

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I don't know which ugly proofs you have seen, but the basic idea is bound to be Sylow's theorem. The proof that immediately comes to mind doesn't seem ugly at all to me: wlog. let $p$ be the biggest of the primes. The number of Sylow $p$-subgroups is $\equiv 1\pmod p$ and divides $qr$. So either it is 1, in which case you are done, or it is $qr$ (it can't be $q$ or $r$, because that would be smaller than $p$).

In the latter case you obviously have a very strong restriction: $(p-1)qr$ elements of order $p$, which leaves no space for multiple Sylow $q$ or $r$-subgroups.

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Assume that $G$ is a simple group with $p and show that $n_r = pq$, $n_q \geq r$ and $n_p \geq q$. Then count the elements to get a contradiction.