Results 1 to 4 of 4

Math Help - Abelian Group Question

  1. #1
    Super Member
    Joined
    Apr 2009
    Posts
    677

    Abelian Group Question

    G is an abelian group.

    a,b are in G
    order of a = m
    order of b = n

    what is this order of ab ?

    Struggling with this. I suspect it is lcm(m,n) but unable to prove it.

    Any help/pointers please?
    Last edited by aman_cc; October 17th 2009 at 05:28 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by aman_cc View Post
    G is an abelian group.

    a,b are in G
    order of a = m
    order of b = n

    what is this order of ab ?

    Struggling with this. I suspect it is lcm(m,n) but unable to prove it.

    Any help/pointers please?
    it's not necessarily lcm(m,n). the only thing we can say is that o(ab) divides lcm(m,n). the point is that o(ab) doesn't only depend on o(a) and o(b). it also depends on the relationship between a

    and b. for example if b=a^{-1}, then o(a)=o(b)=m but o(ab)=1.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Apr 2009
    Posts
    677
    Quote Originally Posted by NonCommAlg View Post
    it's not necessarily lcm(m,n). the only thing we can say is that o(ab) divides lcm(m,n). the point is that o(ab) doesn't only depend on o(a) and o(b). it also depends on the relationship between a

    and b. for example if b=a^{-1}, then o(a)=o(b)=m but o(ab)=1.
    Thanks! I get it.

    I had worked out the following some time back
    <br />
\frac{mn}{\gcd(m,n)^2}|o(ab)

    Does it make sense?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by aman_cc View Post
    Thanks! I get it.

    I had worked out the following some time back
    <br />
\frac{mn}{\gcd(m,n)^2}|o(ab)

    Does it make sense?
    that's correct. so we have \frac{\text{lcm}(m,n)}{\gcd(m,n)} \mid o(ab) \mid \text{lcm}(m,n). this is not bad but to find o(ab), as i mentioned, we need to know how a,b are related.
    Last edited by NonCommAlg; October 17th 2009 at 08:52 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. abelian group
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 16th 2010, 04:41 PM
  2. Is the subgroup of an abelian group always abelian?
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: December 6th 2009, 11:38 PM
  3. Abelian Group
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 16th 2009, 07:14 PM
  4. Abelian p-group
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 6th 2008, 01:34 AM
  5. Is that an abelian group?
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: January 26th 2008, 05:08 AM

Search Tags


/mathhelpforum @mathhelpforum