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Homework Help: Isomorphism between groups, direct product, lcm, and gcd

  1. Jan 25, 2009 #1
    1. The problem statement, all variables and given/known data

    Let a,b, be positive integers, and let d=gcd(a,b) and m=lcm(a,b). Show ZaXZb isomorphic to ZdXZm

    2. Relevant equations

    m=lcm(a,b) implies a|m, b|m and if a,b|c then m|c.
    d=gcd(a,b) implies d|a, d|b and if c|a and c|b then d|c

    let n be an integer with prime decomposition n=p1a1...pkak
    then Zn=Zp1a1X...XZp1ak
    3. The attempt at a solution

    It is clear that d|m

    I considered the prime decomposition of a and b as sets, denote as A and B respectively, and consider the intersection, call this set P. intersection(A,B) = P

    Call the product of the elements in this set P'. I claim P' = gcd(a,b), this should be clear.

    Then I considered the lcm as the product of the elements in A-P and B-P call these sets A' and B' respectively.

    Now we can write the d = lcm(a,b) =A'*B'*P' and this should be clear.

    So... now I want to use the isomorphism as described as in the section above, but I don't know what to do. I think I should use ZaXZb is isomorphic to something like
    (*)
    Zp1k1X...XZpikiXZq1l1X...XZpnln.

    where p are the primes of a and q are the primes of b.

    Great and now I want to show that that ZdXZm is isomorphic to the same thing, but I'm starting to think that it is not.

    I need to make an argument stronger that one of just cardinality, but I don't know what. I'm feel that I need to justify stuff in (*) but I'm not even sure if it is true!

    It took me a long time to think of this and I really need to get it done... help would be greatly appreciated.
     
  2. jcsd
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