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I'm working on the following problem: Let m, n be positive integers. Define addition on Zm ×Zn by (a, b)+(x, y) = (a+x, b+y). Show that Zmn and Zm × Zn are isomorphic if and only if m and n are coprime (i.e. gcd(m, n) = 1).

I can't understand how to define a map from Zmn to Zm x Zn as in Zmn you take a pair of integers and add them together mod mn while in Zm x Zn you have to take two pairs of integers. Can someone help me make sense of this?

Kristina
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    Hint: The Chinese Remainder Theorem gives you a map from $Z_m \times Z_n$ to $Z_{mn}$. Now you have to show that this map is a group isomorphism. For the "only if" part suppose $m$ and $n$ are not co-prime - what is the maximum order of an element of $Z_m \times Z_n$ ? What is the order of $1$ in $Z_{mn}$ ? – gandalf61 May 03 '18 at 15:11

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