If $G$ is isomorphic to the direct product of its subgroups $H$ and $K$, are $H$ and $K$ normal in $G$?

Question

Is this proposition true? Or give a counter example? It comes from this question: if $H$ is a direct factor of $K$ and $K$ is a direct factor of $G$, then $H$ is normal in $G$. If the proposition is true then we are done.If not then how to prove it ..?

Answer 1

Take $G=S_3\times \Bbb{Z}_2$
Since $|((12),0)|=2$, $\langle((12),0)\rangle \cong \Bbb{Z}_2$.
Write $H=S_3\times0$ and $K=\langle((12),0)\rangle$.
Then $H,K$ are subgroups of $G$ where $G\cong H\times K$ since $H\cong S_3$ and $K\cong\Bbb{Z}_2$.
But $K$ is not a normal subgroup of $G$ since $((23),0)((12),0)((23),0)^{-1}=((13),0)\not\in K$.

Answer 2

Take any group $H$ with a non-normal subgroup $K\subseteq H$ and consider $G:=H\times K$.

Now let $H':=H\times\{e\}$ and let $K':=K\times\{e\}$. Both are subgroups of $G$ but $K'$ is not normal. However $K'$ is isomorphic to $K$ and $H'$ is isomorphic to $H$ and thus $G\simeq H'\times K'$.