I have been asked to give all the non-isomorphic, abelian groups with order $12$ & order $15$ respectively.
The answer reads
Order $12$: $\Bbb Z_{12}$ and $\Bbb Z_6 \times\Bbb Z_2$
Order $15$: $\Bbb Z_{15}$
I don't really understand what makes it so clear to see that such groups are non-isomorphic. For example:
Q1) What property makes $\Bbb Z_6 \times \Bbb Z_2$ non isomorphic, but not, say, $\Bbb Z_4 \times\Bbb Z_3$?
Q2) Why is it that for order $15$, we can't have $\Bbb Z_3 \times\Bbb Z_5$? What makes this product isomorphic?
Any help appreciated!
If $m,n\in\Bbb N\setminus\{1\}$ are coprime, then $\Bbb Z_{mn}\simeq\Bbb Z_m\times\Bbb Z_n$. Therefore, $\Bbb Z_4\times\Bbb Z_3\simeq\Bbb Z_{12}$ and $\Bbb Z_5\times\Bbb Z_3\simeq\Bbb Z_{15}$.
But $\Bbb Z_{12}$ has an element of order $12$, whereas $\Bbb Z_6\times\Bbb Z_2$ has no such element. Therefore $\Bbb Z_{12}\not\simeq\Bbb Z_6\times\Bbb Z_2$.