Results 1 to 3 of 3

Math Help - group of rational numbers

  1. #1
    Newbie
    Joined
    Oct 2009
    Posts
    2

    group of rational numbers

    How we can prove additive group of rational numbers (Q,+)is not cyclic, but every finitely generated subgroup of (Q,+)is cyclic.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by makenqau1 View Post
    How we can prove additive group of rational numbers (Q,+)is not cyclic, but every finitely generated subgroup of (Q,+)is cyclic.

    This question obviously belongs in Linear and Abstract Algebra and not here.

    Assume Q = <a/b> and check the prime factorization of b. Now show that
    any multiple m*(a/b) of a/b cannot have different primes in the denominator from the ones that appear in the decomposition of b and thus...

    About every fin. gen. sbgp.: check first for a 2-generator group (pretty easy) and then generalize by induction

    Tonio
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Swlabr's Avatar
    Joined
    May 2009
    Posts
    1,176
    Quote Originally Posted by makenqau1 View Post
    How we can prove additive group of rational numbers (Q,+)is not cyclic, but every finitely generated subgroup of (Q,+)is cyclic.
    There is another nice property about this group - every (non-trivial) image is infinite, but every element of it has finite order. Can you prove this?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Rational numbers?
    Posted in the Algebra Forum
    Replies: 4
    Last Post: November 25th 2010, 04:47 AM
  2. Proof: All rational numbers are algebraic numbers
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: September 5th 2010, 10:26 AM
  3. Replies: 2
    Last Post: May 29th 2009, 01:47 PM
  4. Replies: 8
    Last Post: September 15th 2008, 04:33 PM
  5. Rational Numbers
    Posted in the Calculus Forum
    Replies: 1
    Last Post: July 14th 2007, 03:35 PM

Search Tags


/mathhelpforum @mathhelpforum