Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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
up vote 7 down vote accepted

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.

share|cite|improve this answer
Thanks for your help. – Ester Jul 10 '12 at 3:08
You are welcome. In this very concrete situation, we can really get our hands on the objects, with little machinery needed. – André Nicolas Jul 10 '12 at 3:10

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.