# Centralizer

• October 8th 2011, 12:10 PM
dwsmith
Centralizer
Prove that $C_G(A)=\{g\in G:g^{-1}ag=a, \forall a\in A\}$

Is this just asking me to prove $g^{-1}ag$ iff ga = ag?
• October 8th 2011, 12:23 PM
Drexel28
Re: Centralizer
Quote:

Originally Posted by dwsmith
Prove that $C_G(A)=\{g\in G:g^{-1}ag=a, \forall a\in A\}$

Is this just asking me to prove $g^{-1}ag$ iff ga = ag?

Presumably you are asked to prove that $\left\{g\in G:ga=ag,\;\forall a\in A\right\}=\left\{g\in G:g^{-1}ag=a,\; \forall a\in A\right\}$. Then, yes, it follows from the fact that $ag=ga$ if and only if $g^{-1}ag=a$.