Results 1 to 2 of 2
Like Tree1Thanks
  • 1 Post By TheEmptySet

Thread: Left and Right Cosets

  1. #1
    Super Member
    Joined
    Feb 2008
    Posts
    535

    Left and Right Cosets

    Let G be a group and H a subgroup of G. Let a, b in G. Prove the following:

    If aH = Ha and bH = Hb, then (ab)H = H(ab).

    Can someone show this please? Thanks a lot in advance!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78

    Re: Left and Right Cosets

    Quote Originally Posted by jzellt View Post
    Let G be a group and H a subgroup of G. Let a, b in G. Prove the following:

    If aH = Ha and bH = Hb, then (ab)H = H(ab).

    Can someone show this please? Thanks a lot in advance!
    So we want to show that $\displaystyle (ab)H \subset H(ab)$

    let $\displaystyle k \in (ab)H$

    Then $\displaystyle k=(ab)h$ for some $\displaystyle h \in H$

    By the relation $\displaystyle bH=Hb$ we know that there exists an $\displaystyle h_1 \in H$ such that $\displaystyle bh=h_1b$

    and we know that by $\displaystyle aH=Ha$ there exists an $\displaystyle h_2 \in H$ such that $\displaystyle ah_1=h_2a$

    Putting all of these together we get

    $\displaystyle k=(ab)h=a(bh)=a(h_1b)=(ah_1)b=h_2(ab) \in H(ab)$

    Now you just need to show or justify the other direction (it is exactly the same.)
    Thanks from jzellt
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Left and Right cosets.
    Posted in the Advanced Algebra Forum
    Replies: 12
    Last Post: Feb 2nd 2010, 01:35 PM
  2. Left cosets
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: Jan 11th 2010, 09:43 AM
  3. left and right cosets
    Posted in the Advanced Algebra Forum
    Replies: 10
    Last Post: Nov 4th 2009, 08:42 PM
  4. Left and Right Cosets
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Apr 22nd 2009, 11:06 PM
  5. right and left cosets
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Dec 15th 2008, 06:08 PM

Search Tags


/mathhelpforum @mathhelpforum