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Algebra problem

  1. Oct 26, 2006 #1

    0rthodontist

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    This assignment has not yet been turned in, so I do not want any unfair help on this whatsoever. Please just give me a "yes" or a "no." Do not explain.

    The question:
    Let G be a group acting on itself by conjugation. Show that if a and b are conjugates in G, then the centralizers C(a) and C(b) are equal iff these centralizers are normal subgroups of G.

    My problem:
    I got the normal -> equal direction. But a is a conjugate of itself, and C(a) = C(a), so the other direction would imply that any centralizer is normal. I may be so tired I'm blind, but is this a flawed question?
     
  2. jcsd
  3. Oct 26, 2006 #2

    0rthodontist

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    I've handed the homework in. The professor said that the problem should have said that a and b are distinct.
     
  4. Oct 31, 2006 #3

    0rthodontist

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    The professor talked about this problem today: he had found a counterexample in S6 to the revised proposition stipulating that a and b are distinct. Apparently the other direction should have read something like "if a is in G, and for every b that is a conjugate of a, C(a) = C(b), then C(a) is normal in G."
     
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