Results 1 to 4 of 4

Math Help - Finite subgroup of an infinite Group

  1. #1
    MHF Contributor Amer's Avatar
    Joined
    May 2009
    From
    Jordan
    Posts
    1,093

    Finite subgroup of an infinite Group

    The only subgroup of finite order of a an infinite group is the identity
    My proof

    Let G be a group with infinite order
    let H be a subgroup and let x \in H and x \ne 1
    \{x , x^2 , x^3 , x^4 , ... \} \subseteq H
    which make H with an infinite order
    am I rite ?
    is there any counter example
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45

    Re: Finite subgroup of an infinite Group

    Quote Originally Posted by Amer View Post
    The only subgroup of finite order of a an infinite group is the identity
    Counterexample: G=\mathbb{R}-\{0\} with the usual product and H=\{1,-1\} .
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21

    Re: Finite subgroup of an infinite Group

    More generally, take any field k then if there exists non-zero x\in k with k^n=1 (i.e. a non-trivial n^{\text{th}} root of unity) then the group \mu_n of n^{\text{th}}-roots of unity form a finite subgroup of k^\times. This clearly generalizes Dr. Revilla's example (so that, for example, the 3-roots of unity in \mathbb{C}).


    But, perhaps even more of a trivial counterexample. Take any infinite group G and consider G\times\mathbb{Z}_2....
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Amer's Avatar
    Joined
    May 2009
    From
    Jordan
    Posts
    1,093

    Re: Finite subgroup of an infinite Group

    Thanks
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 7
    Last Post: February 19th 2011, 04:29 PM
  2. Centralizer of a subgroup is a subgroup of the main group
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 1st 2010, 08:16 AM
  3. Finite group with identity subgroup.
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 18th 2010, 10:17 AM
  4. Finite group with identity subgroup.
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: February 18th 2010, 06:25 AM
  5. Finite subgroup of Q/Z
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 8th 2008, 03:29 AM

Search Tags


/mathhelpforum @mathhelpforum