Results 1 to 2 of 2

Math Help - Finite subgroups of Q/Z

  1. #1
    Junior Member
    Joined
    Nov 2012
    From
    Taiwan
    Posts
    41

    Post Finite subgroups of Q/Z

    How do I prove that every finite subgroup of Q/Z is cyclic?
    Could someone help me?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,392
    Thanks
    759

    Re: Finite subgroups of Q/Z

    let G be a finite subgroup of Q/Z. since G is finite, it is finitely generated by some finite set of cosets {qi+Z: i = 1,2,...,n}.

    suppose we pick each qi in (0,1) and write qi = ai/bi, with gcd(ai,bi) = 1, for each i.

    let m = lcm(b1,...,bn).

    we then have that G is contained in the cyclic group <(1/m)+Z>, and thus, being a subgroup of a cyclic group, is itself cyclic.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. groups, subgroups and the finite hypothesis
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: October 18th 2012, 08:55 AM
  2. normal p-subgroups of a finite group
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: August 17th 2012, 03:21 PM
  3. [SOLVED] Cyclic Subgroups of a finite group G
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 10th 2011, 10:05 AM
  4. subgroups of finite cyclic groups
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: August 30th 2009, 05:47 PM
  5. Finite size if finite number of subgroups
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: March 9th 2006, 12:32 PM

Search Tags


/mathhelpforum @mathhelpforum