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Let f:G -> H be a surjective homomorphism. |C_G(g)| >= |C_H(f(g)|

  1. May 22, 2013 #1
    Let f:G --> H be a surjective homomorphism. |C_G(g)| >= |C_H(f(g)|

    1. The problem statement, all variables and given/known data

    Suppose G is a finite group and H is a group, where θ:G→H is a surjective homomorphism. Let g be in G. Show that |CG(g)| ≥ |CH(θ(g))|.

    2. Relevant equations

    This problem has been bugging me for a day now. I'm studying for my qualifying exam and doing very well otherwise. I sure could use some peace of mind though concerning this problem. I tend to obsess over things I can't figure out.

    3. The attempt at a solution

    Obviously, θ[CG(g)] ≤ CH(θ(g)), so CG(g) is contained in the pullback of CH(θ(g)). Beyond that, I'm stuck and I would greatly appreciate assistance.
     
  2. jcsd
  3. May 25, 2013 #2
  4. May 26, 2013 #3
    It wasn't me, but thanks for the link!
     
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