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

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 ..?

• Using the given information can you define a map from $G$ to some different group $G^{'}$ whose kernel is $H$. – math Feb 14 at 9:55

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 $$((23),0)((12),0)((23),0)^{-1}=((13),0)\not\in K$$.
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 $$G\simeq H'\times K'$$.