We consider the two abelian groups
$\Bbb Z_4$ with addition modulo $4$.
$\Bbb Z_2 \times\Bbb Z_2$ with component-by-component addition modulo $2$.
a) Show there is no isomorphism between these groups.
b) Show that this groups are the only abelian groups( except isomorphism) with $4$ elements.
So I was able to solve a): all elements in $\Bbb Z_2 \times\Bbb Z_2$ have order $ \leq 2 $, but $1$ in $\Bbb Z_4$ have order $4$.
But how can I solve b) ? Do you have a hint? Maybe I can use a)?
Best Answer
Actually, any group of order $4$ is abelian. Then if you're willing to use a sledgehammer, you could use the fundamental theorem of finite abelian groups.
Or by Lagrange, either there are three elements of order two, or else it's cyclic.