Showing a non-isomorphism of groups

Question

I need to show that $\Bbb Z^*_8$ is not isomorphic to $\Bbb Z^*_{10}$.

$\Bbb Z^*_n$ means integers up to $n$ coprime with $n$

I do not know how to do this. I have difficulties doing proofs involving isomorphisms. A methodological answer would be highly appreciated.

Thanks in advance!

Answer 1

Order of $3$ in $\mathbb Z_{10}^*$ is $4$, while the order of elements from $\mathbb Z_8^*$ is $2$.

Answer 2

Hint: see if you can show that one of these groups is cyclic, whereas the other is not.