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I am wondering how do we find all subgroups of a direct product? Is there a method to find it?

For example, how can we find all the subgroups of $\mathbb{Z}_2\times\mathbb{Z}_2$? There is the answer: $\{(0, 0), (1, 0)\}, \{(0, 0), (0, 1)\}, \{(0, 0), (1, 1)\}$, and the improper and trivial subgroup. But I do not know the method how to find it.

All I can think about is the following: since the order of $\mathbb{Z}_2\times\mathbb{Z}_2$ is $4$, the order of a subgroup of it can be $1,2$ or $4$.

If order is $1$, then it is trivial, and if order is $4$, it is proper. Then, the order of a subgroup is $2$. We know all the elements of $\mathbb{Z}_2\times\mathbb{Z}_2$, so we try all the pairs.

Is that the method?

But what if we are given a bigger order product, such as $\mathbb{Z}_2\times\mathbb{Z}_8\times\mathbb{Z}_3\times\mathbb{Z}_9$? It looks very long to try all possibilities.

Can anyone help?


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@julien, actually i did not copy and paste, i tried to use mathematical notations as much as i could – bigO Apr 14 '13 at 20:25
Ah, sorry then! But here is something you might find interesting. – 1015 Apr 14 '13 at 20:26
Thank you, i will have a look at it – bigO Apr 14 '13 at 20:26
For just the normal subgroups you can take a look at… (all your examples are abelian, so all subgroups are normal then). – j.p. Apr 15 '13 at 12:26
up vote 4 down vote accepted

I don't think that you can find a nice method. It is a difficult problem even for subdirect sums, as L.Fuchs wrote (Laszlo Fuchs, Infinite Abelian Groups, vol.1, Sec.II.8, p.42):

"There are a great number of subdirect sums in a direct product of groups, and no complete survey of subdirect sums is known except for the case of subdirect sums of two groups."

After this quote L.Fuchs described a solution for the subdirect sum of two Abelian groups.

Addendum: Thus to find all subgroups of $G\times H$, you may find all subgroups $A\in G$ and $B\in H$, and then all subdirect sums of $A\times B$.

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