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Let $G_1$ and $G_2$ be subgroups of a group $G$.

Assume that $G$ is isomorphic to $G_1\times G_2$.

Then is it necessary that $G_1$ and $G_2$ are normal in $G$?

Clealry $G_1\cong G_1\times \{e\}\trianglelefteq G_1\times G_2$ and $G_2\cong \{e\}\times G_2\trianglelefteq G_1\times G_2$.

So it is easy to infer that $G$ contains normal subgroups $H_1$ and $H_2$ such that $H_1\cong G_1$ and $H_2\cong G_2$. But does that force $G_1,G_2\trianglelefteq G$?

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Let $H$ be a non-normal subgroup of $G_1$ and $G_2\cong H$. Then $G=G_1\times G_2$ is isomorphic to $G_1\times H$ yet $H$ is not normal in $G$.

Edit: If the isomorphism $G_1\times G_2\to G$ is given by $(g_1,g_2)\mapsto g_1g_2$, then the answer is yes, $G_1$ is normal in $G$.

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  • $\begingroup$ I assume you mean "$H$ is a non-normal subgroup of $G$ (rather than $G_1$)..." How are we sure that such a subgroup of $G$ exists? $\endgroup$ – caffeinemachine Aug 19 '14 at 16:53
  • $\begingroup$ No, I meant $G_1$. Explicit example: $G_1$ is the symmetric group $S_3$, $H=\langle (1\,2)\rangle\cong \Bbb Z/2$ and $G=S_3\times \Bbb Z/2$. $\endgroup$ – Quang Hoang Aug 19 '14 at 16:54
  • $\begingroup$ I could not follow your counterexample. Can you please mentions what are $G$, $G_1$ and $G_2$? $\endgroup$ – caffeinemachine Aug 19 '14 at 16:59
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    $\begingroup$ $G=S_3\times \Bbb Z/2$, $G_1=S_3$ and $G_2=\langle (1\,2)\rangle$ sitting inside $S_3$. $\endgroup$ – Quang Hoang Aug 19 '14 at 17:00
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    $\begingroup$ To be very precise, I believe you intended $G=S_3\times \mathbb Z/2$, $G_1=S_3\times \{e\}$, $G_2=\langle(12)\rangle\times \{e\}$, where $e$ is the identity of $\mathbb Z/2$. Thanks. $\endgroup$ – caffeinemachine Aug 19 '14 at 17:05

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