Results 1 to 2 of 2

Math Help - generators and relations

  1. #1
    Super Member
    Joined
    Mar 2010
    Posts
    979
    Thanks
    236

    generators and relations

    Here is the problem I'm working on, from an old qualifying exam:

    Let M be the \mathbb{Z}-module generated by a, b, c with the relations 4a+3b+3c=2a-b+3c=0. Express M as a direct sum of cyclic modules. What are the orders of these modules?

    There's only two relations, so I'm guessing that one of the modules is going to be \mathbb{Z}. I managed to find a vector that is perpendicular to (4,3,3) and (2,-1,3), which is (6, -3, -5). I'm not sure what to do with it though, or if I'm even going in the right direction.

    Thanks,
    Hollywood
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Mar 2010
    Posts
    979
    Thanks
    236

    Re: generators and relations

    I think I've made some progress on this problem.

    4a+3b+3c=0
    2a-b+3c=0

    Subtracting twice the second from the first gives:

    2a-b+3c=0
    5b-3c=0

    And now subtracting the second from the first gives:

    2a-6b+6c=0
    5b-3c=0

    Let e=a-3b+3c, so a=e+3b-3c gives:

    2e=0
    5b-3c=0

    The first equation gives \mathbb{Z}/2\mathbb{Z}, and for the second, my reasoning is that \mathbb{Z}\times\mathbb{Z} gets divided into cosets of \{(3x,5x):x\in\mathbb{Z}\}, and so that gives a factor of \mathbb{Z}. So the answer is M\cong\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}.

    Is that correct?

    - Hollywood
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Generators for Z/nZ
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: October 15th 2011, 04:18 PM
  2. [SOLVED] s and sr generators for D2n
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 3rd 2011, 03:52 AM
  3. [SOLVED] Generators and relations
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: October 3rd 2011, 12:16 AM
  4. Replies: 1
    Last Post: September 19th 2011, 01:09 PM
  5. Set of Generators and Relations
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: September 6th 2011, 01:40 AM

Search Tags


/mathhelpforum @mathhelpforum