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Prove:

a) $\mathbb{Z}_{2} \times \mathbb{Z}_2 \ncong \mathbb{Z}_4$

b) $\mathbb{Z}_2 \times \mathbb{Z}_5 \cong \mathbb{Z}_{10}$

For a) I figured out that in $\mathbb{Z}_2 \times \mathbb{Z}_2$ every element is its own inverse and in $\mathbb{Z}_4$ that is not the case and I'm not sure if that is enough to show they aren't isomorphic.

For b) I don't know how to find isomorphism.

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    $\begingroup$ Welcome to Mathematics Stack Exchange. For a), that is enough. For b), map $(1,0)$ to $5$ and $(0,1)$ to $2$ $\endgroup$ Dec 27, 2019 at 14:53
  • $\begingroup$ For (a). Yes, of course it is enough to show that they are not isomorphic. $\endgroup$
    – almagest
    Dec 27, 2019 at 15:07
  • $\begingroup$ Please ask only one question per post. Having multiple questions in the same post is discouraged and such posts may be put on hold, see meta. $\endgroup$
    – Shaun
    Dec 27, 2019 at 15:25
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    $\begingroup$ @Shaun I will keep that in mind, thanks $\endgroup$ Dec 27, 2019 at 15:35
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    $\begingroup$ you need to map $(1,0)$ to something of order $2$ in $\mathbb Z_{10}$, which is $5$; you could map $(0,1)$ to anything of order $5$ in $\mathbb Z_{10}: 2, 4, 6, $ or $8$ $\endgroup$ Dec 27, 2019 at 15:38

1 Answer 1

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There's a theorem stating that $C_m \times C_n$ is cyclic (and then isomorphic to $C_{mn}$) if and only if $\operatorname{gcd}(m,n)=1$.

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