When is $\mathbb{Z}_m\times\mathbb{Z}_n$ isomorphic to $\mathbb{Z}_{mn}$?

Asked Jun 08 '15 at 06:13

Active Jun 08 '15 at 06:18

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Prove that $\mathbb{Z}_m\times\mathbb{Z}_n$ is isomorphic to $\mathbb{Z}_{mn}$ whenever $m$ and $n$ are coprime.

$\mathbb{Z}_m$ represents a cyclic group of order $m$.

I have no knowledge of rings and fields. I only know group theory.

In general, I would like to know how to prove that $2$ groups are isomorphic to each other? Is the only way to solve it is to show a bijective homomorphism?

edited Jun 08 '15 at 06:18

Zev Chonoles

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asked Jun 08 '15 at 06:13

idpd15

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see also Prove that $\mathbb Z_{m}\times\mathbb Z_{n} \cong \mathbb Z_{mn}$ implies $\gcd(m,n)=1$. – Zev Chonoles Jun 08 '15 at 06:19
If you have a textbook, this fact is commonly exploited to prove the Chinese Remainder Theorem, and is likely proved there. – Jun 08 '15 at 06:22
http://math.stackexchange.com/questions/375348/prove-that-mathbb-z-m-times-mathbb-z-n-cong-mathbb-z-mn-implies-gcd This also helps. – idpd15 Jun 08 '15 at 06:29

When is $\mathbb{Z}_m\times\mathbb{Z}_n$ isomorphic to $\mathbb{Z}_{mn}$?

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