1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?



Similar Discussions: Isomorphism between groups, direct product, lcm, and gcd
  1. Group theory (Replies: 0)

  2. Product measures (Replies: 0)

Loading...