Questioning Centralizers: A Group Acting On Itself

[0rthodontist]

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?

 

[0rthodontist]

I've handed the homework in. The professor said that the problem should have said that a and b are distinct.

 

[0rthodontist]

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."