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Let f:G --> H be a surjective homomorphism. |C_G(g)| >= |C_H(f(g)|
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))|.
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.
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.
Homework Statement
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))|.
Homework 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.
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.