When Is $HK$ A Subgroup Of $G$

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Problem (When Is $HK$ A Subgroup Of $G$)

Suppose $H$ and $K$ are subgroups of $G$.

1. Prove that $HK$ is a subgroup of $G$ if and only if $HK=KH$.
2. Prove that if either $H$ or $K$ is normal in $G$, then $HK$ is a subgroup of $G$.

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