What are the values of $a$ and $b$ so that $\Bbb Z_2\times\Bbb Z_3\times\Bbb Z_4\times\Bbb Z_9$ is isomorphic with $\Bbb Z_a\times\Bbb Z_b$?
I'm studying group theory, (at basic level), and i got this problem:
Find all pairs $(a,b)$ of positive integers so that:
$$\Bbb Z_2\times\Bbb Z_3\times\Bbb Z_4\times\Bbb Z_9$$
is isomorphic with
$$\Bbb Z_a\times\Bbb Z_b.$$
I know that $\Bbb Z_2\times\Bbb Z_3$ is cyclic, so is isomorphic with $\Bbb Z_6$, and so $\Bbb Z_9\times\Bbb Z_4$ with $\Bbb Z_{36}$ , but in this way I just found two isomorphisms, $\Bbb Z_6\times\Bbb Z_{36}$ and $\Bbb Z_{18}\times\Bbb Z_{12}$. Is it right?
Thank you so much.
Yes, that's right. It follows from coprimality and the Chinese remainder theorem.