Are Abelian Groups of Relatively Prime Orders Isomorphic?

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SUMMARY

Abelian groups of relatively prime orders m and n are isomorphic to the product of their respective cyclic groups. Specifically, if m and n are relatively prime positive integers, then the number of abelian groups of order mn is the product of the number of abelian groups of orders m and n, denoted as rs. This conclusion is supported by the fact that the greatest common divisor gcd(m, n) equals 1, leading to the isomorphism C_{mn} ≅ C_{m} × C_{n}.

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Homework Statement



Let m and n be relatively prime positive integers. Show that if there are, up to isomorphism, r abelian groups of order m and s of order n, then there are rs abelian groups of order mn.

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The Attempt at a Solution



I'm not sure how to go about this. I was thinking of saying that since m and n are relatively prime, the gcd(m,n)=1; wouldn't this then imply that the group order would be mn? Because mn is the lcm of m and n?

Any help is appreciated.
 
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\hbox{gcd}(m,n)=1 \Rightarrow C_{mn}\cong C_{m} \times C_{n}.

HINT:1. If C_m=<a>, C_n=<b> then prove that C_{m} \times C_{n}=<d> where d=(a,b)
2. Cyclic groups same orders are isomorphic.
 
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