Results 1 to 4 of 4
Like Tree1Thanks
  • 1 Post By Deveno

Math Help - Cosets

  1. #1
    Newbie
    Joined
    Sep 2012
    From
    United States
    Posts
    12

    Cosets

    1. For each g in G, show that y is an element of xZ(g) (left coset of the centralizer) if and only if ygy-1 = xgx-1

    2. Show that y is an element of xZ(G) (left coset of the center) if and only if ygy-1 = xgx-1 for all g in G.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Sep 2012
    From
    Washington DC USA
    Posts
    525
    Thanks
    147

    Re: Cosets

    1) Do you know the definition of the centralizer of g?
    2) Do you know the definition of the center of a group?
    3) Do you know the definition of a coset?
    These two problems are just straightforward consequences of the definitions.
    Maybe this will help: a is in bH (where H is a subgroup of some group G, a and b are elements of G) if and only if (b-inverse)(a) is in H.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2012
    From
    United States
    Posts
    12

    Re: Cosets

    I know the definitions, I'm just having trouble seeing how to use them to show that those statements are true.

    The centralizer of g in G is the set of elements of G which commute with g. The center of a group is the set of elements that commute with every element of G. A coset is basically a shifted subgroup of G, more specifically gH = {gh: h E G} where H is a subgroup of G.

    I know how to find centralizers, centers, and cosets of a group, but I'm not really sure how to show these proofs.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,401
    Thanks
    762

    Re: Cosets

    this can be rephrased as follows:

    different conjugates of g give rise to distinct cosets of Z(g). think about what this may mean. anyway:

    suppose y is in xZ(g). what does this mean? it means x-1y is in Z(g). so x-1y commutes with g:

    x-1yg = gx-1y
    x-1ygy-1 = gx-1 (multiplying on the right by y-1)
    ygy-1 = xgx-1 (multiplying on the left by x).

    that's what we wanted to prove for y in xZ(g) implies ygy-1 = xgx-1.

    the other implication merely runs this argument backwards.

    that's part (1) of your problem.

    part (2) is an easy generalization. for if x-1y commutes with every element g of G, it surely lies in Z(G).
    Thanks from TheHowlingLung
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Cosets
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: November 6th 2010, 05:26 PM
  2. cosets
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: October 27th 2009, 04:51 PM
  3. Cosets
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: September 22nd 2009, 03:40 PM
  4. Cosets
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 9th 2008, 04:22 PM
  5. cosets
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: October 8th 2008, 12:23 AM

Search Tags


/mathhelpforum @mathhelpforum