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

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In summary, the problem states that given a finite group G and a group H, where θ:G→H is a surjective homomorphism, the number of elements in the centralizer of g in G is greater than or equal to the number of elements in the centralizer of θ(g) in H.
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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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It wasn't me, but thanks for the link!
 

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

What is a surjective homomorphism?

A surjective homomorphism is a function between two groups, G and H, that preserves the group operation and maps every element in G to an element in H. It is also known as an onto homomorphism, meaning that every element in H has at least one preimage in G.

What does the notation "f:G -> H" mean in this context?

The notation "f:G -> H" indicates that the function f maps elements from the group G to the group H.

What is the significance of the equation |C_G(g)| >= |C_H(f(g)|?

This equation represents a property of surjective homomorphisms known as the "counting theorem." It states that the size of the centralizer of an element g in G is always greater than or equal to the size of the centralizer of its image f(g) in H.

How is the counting theorem useful in studying surjective homomorphisms?

The counting theorem allows us to compare the sizes of centralizers in different groups, which can provide insights into the structure and relationships between the groups. It also helps in proving other theorems and properties related to surjective homomorphisms.

Can the counting theorem be extended to other types of homomorphisms?

Yes, the counting theorem can be extended to other types of homomorphisms, such as injective and bijective homomorphisms. However, the inequality may not always hold in these cases, and there may be additional conditions that need to be met.

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