I am looking for a simple counter example to prove that if $H$ is a subgroup and $K$ is a normal subgroup of $G$, $HK$ is not necessarily a normal subgroup of $G$, though it would always be a subgroup.
Thanks in advance.
2026-03-29 19:30:33.1774812633
If $H$ is a subgroup and $K$ is a normal subgroup of $G$ then $HK$ is a subgroup of $G$ but it need not be normal
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Take $K=\{e\}$ and $H$ as any non-normal subgroup of $G$.