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I need to prove the group of the units of $\mathbb{Z}_3\times\mathbb{Z}_3$ is isomorphic to the Klein-4 group. But I'm really struggling to prove this. Any hints to start me off in the right direction??

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List the units of $\mathbb{Z}_3\times\mathbb{Z}_3$. Whot you have? – MathOverview Nov 16 '12 at 16:38
First step: have you determined which elements are units, and this how many units there are? – Derek Allums Nov 16 '12 at 16:39
up vote 4 down vote accepted

There's really only one thing you can do: list the units of $\mathbb{Z}_3\times\mathbb{Z}_3$, see if there are four of them, and then figure out whether or not they form a cyclic group.

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Start with figuring out the group of units of $\mathbb{Z}_3$. Then, how does the unit group of $\mathbb{Z}_3\times \mathbb{Z}_3$ differ?

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