Results 1 to 2 of 2

Math Help - Centralizer of a subgroup is a subgroup of the main group

  1. #1
    Newbie
    Joined
    Mar 2010
    Posts
    5

    Centralizer of a subgroup is a subgroup of the main group

    Hello! This is my first post here, and I believe I checked all previous posts and could not find one like this.

    The question states:
    If H is a subgroup of G, then by the centralizer C(H) of H we mean the set {x in G | xh=hx for all h in H}. Prove that C(H) is a subgroup of G.

    I have attempted to use the 1 step subgroup test:

    1. C(H)={x in G | xh=hx for all h in H}.
    2. e is in C(H) since he=eh for all h in H, so C(H) is not empty.
    3. Assume a,b is in C(H).
    4. Show ab^-1 is in C(H).
    a in C(H) mean for all h in H, h is in G, ah=ha
    b in C(H) mean for all h in H, h is in G, bh=hb
    I get stuck here, obviously with the majority of the proof to go, the main issue is how do I prove that it is a subgroup of G and not simply that it is a subgroup of H? It makes sense that if it is a subgroup of H, which is a subgroup of G, that it is, but how do I proceed?

    Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,697
    Thanks
    1469
    First, if ah= ha for all h, then, multiplying both sides of that equation by a^{-1} on the left,
    a^{-1}(ah)= h= a^{-1}ha.
    But then, multiplying both sides of that by a^{-1} on the right,
    ha^{-1}= (a^{-1}ha)(a^{-1}= (a^{-1}h.

    That is, if a is in the centralizer, then so is [tex]a^{-1}[tex]

    Of course, if a and b are both in the cetralizer then (ab)h= a(bh)= a(hb)= (ah)b= (ha)b= h(ab).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: March 2nd 2011, 08:07 PM
  2. The Center of a Group as the centralizer of a subgroup.
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: May 26th 2010, 07:05 AM
  3. characterisitic subgroup implies normal subgroup
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 8th 2010, 03:13 PM
  4. Is Centralizer subgroup?
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: September 24th 2008, 04:11 PM
  5. Normal subgroup interset Sylow subgroup
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: May 10th 2008, 12:21 AM

Search Tags


/mathhelpforum @mathhelpforum