Centralizer

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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 $g^{-1}ag=a$ .

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