$H,K$ normal subgroups of a finite group $G$ , $G \cong H \times K$ , every element of $H$ commutes with every element of $K$ , then is $G=HK$?

Question

Let $H,K$ be normal subgroups of a finite group $G$ such that $G$ is isomorphic with $H \times K$ and every element of $H$ commutes with every element of $K$ , then is it necessary that $G=HK$ ? ( Note that I am not assuming $H \cap K=\{e\}$ , because assuming that , since $G$ is finite , the claim would trivially be true ) .If this is not true , what if we also assume that $G$ is abelian ? Please help . Thanks in advance

Answer 1

If you don't assume anything about how $H$, $K$ lie inside $G$, then is this not trivially false. For example take $G=\{1,a,b,ab\}$ be the Klein four, and $H=K=\langle a\rangle$.

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