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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?