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Homework Help: Proof for abelian groups ?

  1. Mar 13, 2012 #1
    1. The problem statement, all variables and given/known data

    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.

    2. Relevant equations

    3. 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.
  2. jcsd
  3. Mar 13, 2012 #2
    [tex]\hbox{gcd}(m,n)=1 \Rightarrow C_{mn}\cong C_{m} \times C_{n}[/tex].

    HINT:1. If [tex]C_m=<a>, C_n=<b>[/tex] then prove that [tex]C_{m} \times C_{n}=<d>[/tex] where [tex]d=(a,b)[/tex]
    2. Cyclic groups same orders are isomorphic.
    Last edited: Mar 13, 2012
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