Suppose $p<q$, where $p,q$ are primes and we have a non-abelian group $G$ of order $p^2q$. Is it true that it has a subgroup which is not normal? I try to use Sylow's theorems. We take Sylow subgroups of order $p^2$ and $q$. They are normal , otherwise we have a contradiction. Now i want to say that that $G$ must be abelian, but don't know why it is true...
2026-03-26 21:26:03.1774560363
Question about a non-abelian group of order $p^2q$
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Hint: suppose that all subgroups were normal. Then $G$ has normal subgroups $M$ of order $p^2$ and $N$ of order $q$. What can you say about (the structure of) those subgroups and what about the order of $MN$?