Results 1 to 2 of 2

Math Help - group theory

  1. #1
    Junior Member
    Joined
    Apr 2009
    Posts
    36

    group theory

    If G is a finite set closed under an associative operation such that ax=ay forces x=y and ua=wa forces u=w, for every a,x,y,u,w in G, prove that G is a group.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by mpryal View Post

    If G is a finite set closed under an associative operation such that ax=ay forces x=y and ua=wa forces u=w, for every a,x,y,u,w in G, prove that G is a group.
    this is a good question! let G=\{x_1, \cdots , x_n \}. then G=\{x_1x_i: \ 1 \leq i \leq n \} by the left cancellation property and so x_1x_{i_0}=x_1, for some 1 \leq i_0 \leq n. let x_{i_0}x_1=x_j. then by associativity we

    have: x_1^2=x_1x_{i_0}x_1=x_1x_j, and thus by the left cancellation property x_1=x_j and hence x_{i_0}x_1=x_1. the claim is that x_{i_0}=1_G. by the right cancellation property: G=\{x_ix_1: \ 1 \leq i \leq n \}.

    thus for any x_j there exists x_k such that x_j=x_kx_1. thus: x_jx_{i_0}=x_k(x_1x_{i_0})=x_kx_1=x_j. now let x_{i_0}x_j=x_r. then x_j^2=x_jx_{i_0}x_j=x_jx_r, and so by the left cancellation property: x_r=x_j.

    thus x_{i_0}x_j=x_j. this proves that x_{i_0}=1_G. so the only thing left is to show that every element of G has an inverse. let x_s \in G. then G=\{x_sx_i: \ 1 \leq i \leq n \}. thus there exists x_t \in G such

    that x_sx_t=1. so every element of G has a right inverse. particularly x_tx_u = 1, for some x_u \in G. but then x_u=x_sx_tx_u=x_s. hence x_tx_s=1 and the proof is complete.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: November 4th 2009, 09:52 AM
  2. Quick questions on Group Theory - Cosets / Normal Group
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: October 16th 2009, 08:39 AM
  3. Group Theory - Sylow Theory and simple groups
    Posted in the Advanced Algebra Forum
    Replies: 16
    Last Post: May 16th 2009, 11:10 AM
  4. Group Theory Question, Dihedral Group
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: March 4th 2008, 10:36 AM
  5. Group Theory
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: July 12th 2007, 09:28 AM

Search Tags


/mathhelpforum @mathhelpforum