# Subgroups of $\Bbb Z_5 \times \Bbb Z_5$

Find all subgroups of $\Bbb Z_5 \times \Bbb Z_5$. I can see that the non-trivial ones are of order $5$. But how do I find them exactly?

Thanks for any help.

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Groups of order 5 are cyclic... –  lhf Jul 10 '12 at 2:46
List the non-trivial ones. First coordinate could be always $0$. In all other cases, everything is known once we know the number $b$ such that $(1,b)$ is in the subgroup. –  André Nicolas Jul 10 '12 at 2:56

We list the subgroups of order $5$. There is the group generated by $(0,1)$. Then there are the groups generated by $(1,b)$, where $b$ is an element of $\mathbb{Z}_5$. That's all. We can if we wish give the addition table for each.