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Abstract Algebra: Commutative Subgroup

  1. Dec 16, 2009 #1
    1. The problem statement, all variables and given/known data

    Let G be a group and let a, b be two fixed elements which commute with each other (ab = ba). Let H = {x in G | axb = bxa}. Prove that H is a subgroup of G.

    2. Relevant equations


    3. The attempt at a solution

    I'm using the subgroup test. I know how to show that the identity of G exists in H and that if x1, x2 exist in H then x1 times x2 exists in H but need help proving that if x is in H then x^-1 is also in H. I've tried starting with axb = bxa and multiplying on the right and left with various combinations of a, x^-1, and b, but can't get it to the form ax^-1b = bx^-1a.

    Please let me know if you have any ideas, thanks.
  2. jcsd
  3. Dec 16, 2009 #2
    Prove that [b-1,a] = [b,a-1] = 0 and use that along with the fact that axb = bxa.
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