# Isomorphism of finite groups [closed]

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.

• Welcome to Mathematics Stack Exchange. For a), that is enough. For b), map $(1,0)$ to $5$ and $(0,1)$ to $2$ Dec 27, 2019 at 14:53
• For (a). Yes, of course it is enough to show that they are not isomorphic. Dec 27, 2019 at 15:07
• 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. Dec 27, 2019 at 15:25
• @Shaun I will keep that in mind, thanks Dec 27, 2019 at 15:35
• 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 $8$ Dec 27, 2019 at 15:38

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