The Product of Distinct Sylow $p$-Subgroups Can Never be a Subgroup
Problem 544
Let $G$ a finite group and let $H$ and $K$ be two distinct Sylow $p$-group, where $p$ is a prime number dividing the order $|G|$ of $G$.
Prove that the product $HK$ can never be a subgroup of the group $G$.
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