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?