Is a subgroup of a product of two groups necessarily a product of two subgroups ?

Question

Is a subgroup of a product of two groups necessarily a product of two subgroups ? Thanks !

Do you mind clarifying your question a little? Is your product a direct product? If my product of two groups is $G \times H$, then are you asking for two subgroups of $G$ and/or $H$? And by saying group $A$ is $B$, do you mean they are isomorphic? — Myridium, Oct 21 '16 at 15:25
@Myridium-Yeah i mean a direct product and i'm asking for two subgroups of G AND H. I'm so sorry because i only study mathematics in french and i never had the chance to define these English terms.Thanks a lot. — Zakaria Oussaad, Oct 21 '16 at 15:33
How's this then? Is a subgroup $S \leq G \times H$ isomorphic to $A \times B$ for some $A \leq G$ and $B \leq H$? — Myridium, Oct 21 '16 at 15:34
$S_3 \times S_3$ has a subgroup of index $2$ which is not isomorphic to a direct product of two subgroups of $S_3$. — Derek Holt, Oct 21 '16 at 15:36
Related if you are talking about isomorphisms : https://math.stackexchange.com/questions/529023 — Watson, Oct 21 '16 at 16:45
Have a look at Goursat's theorem that is generalized in (https://arxiv.org/abs/1109.0024) — Jean Marie, Oct 21 '16 at 17:19
Possible duplicate of Is any subgroup of a direct product isomorphic to a direct product of subgroups? — Moishe Kohan, Oct 21 '16 at 17:52

score 1 · Answer 1 · answered Oct 21 '16 at 15:12

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Not necessarily, Consider the subgroup of $\mathbb Z\times \mathbb Z$ generated by $(1,1)$

answered Oct 21 '16 at 15:12

Asinomás

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This is the product of $\mathbb Z$ with ${1}$. – Myridium Oct 21 '16 at 15:13
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@Myridium what substance are you on? – Asinomás Oct 21 '16 at 15:15
Am I making a dumb mistake? The way I interpret the OP's question is such that in this case, they are looking for a direct product of subgroups of $\mathbb Z$ isomorphic to the subgroup of $\mathbb Z \times \mathbb Z$ generated by $(1,1)$ (which is $\mathbb Z$). – Myridium Oct 21 '16 at 15:17
Oh ok, I guess that makes sense. I'll try to come up with something under that interpretation. – Asinomás Oct 21 '16 at 15:18
What about this subgroup of Z² : {(x, x) / x in Z} ? – Zakaria Oussaad Oct 21 '16 at 15:25
@ZakariaOussaad - That is $\mathbb Z^2$ which is $\mathbb Z \times \mathbb Z$. The only subgroups of $\mathbb Z^2$ are ${1}$, $\mathbb Z$ and $\mathbb Z^2$, so this may not be a good example. – Myridium Oct 21 '16 at 15:27

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