I want to prove that the groups $\Bbb{Z}_2\ \otimes\ \Bbb{Z}_4 $ and $\Bbb Z_8$ are not isomorphic to each other.
I have figured out that the former has $2$ generators, while the latter has just $1$. Thus I should try to prove by contradiction that if a mapping exists between the generators of the former group and some group elements of the latter, there should be something wrong in it, like: $$f(g_1)f(g_2) \neq f(g_1g_2)$$ a property which is highly important since it helps us to preserve the structure when we move to the new group.
But I am not being able to move beyond this. I am new to group theory so excuse me if this is trivial. Any help is appreciated.
Isomorphisms preserve orders of elements.${}^\dagger$ Note that $\Bbb Z_8$ has an element of order $8$, whereas $\Bbb Z_2\otimes \Bbb Z_4$ does not.
$\dagger$ This means that for any group isomorphism $\varphi:G\to H$ and any $g\in G$, we have $|g|=|\varphi (g)|$.