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.