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

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SUMMARY

The discussion focuses on the relationship between the centralizers of elements in finite groups under a surjective homomorphism. Specifically, it establishes that for a surjective homomorphism θ: G → H, the inequality |C_G(g)| ≥ |C_H(θ(g))| holds true. Participants emphasize the importance of understanding the containment of centralizers, where C_G(g) is contained within the pullback of C_H(θ(g)). This foundational result is crucial for those preparing for advanced studies in group theory.

PREREQUISITES
  • Understanding of group theory concepts, specifically finite groups.
  • Familiarity with homomorphisms, particularly surjective homomorphisms.
  • Knowledge of centralizers in group theory, denoted as C_G(g) and C_H(θ(g)).
  • Basic proficiency in mathematical proofs and inequalities.
NEXT STEPS
  • Study the properties of surjective homomorphisms in group theory.
  • Learn about the structure and significance of centralizers in finite groups.
  • Explore the implications of the pullback of centralizers in homomorphic mappings.
  • Review advanced topics in group theory, such as Sylow theorems and group actions.
USEFUL FOR

This discussion is beneficial for graduate students in mathematics, particularly those specializing in abstract algebra and group theory, as well as educators and researchers looking to deepen their understanding of homomorphisms and centralizers.

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

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.
 
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