Exercise 2.2.1 ($C_G(A) = \{g \in G \mid g^{-1} a g = a ext{ for all } a \in A\}$)

Question

Prove that $C_G(A) = \{g \in G \mid g^{-1} a g = a 	ext{ for all } a \in A\}.$

Richard Ganaye · Accepted Answer

\begin{proof} 
Note that for all $g \in G$ and for all $a \in A$,
\begin{align*}
ga g^{-1} = a &\iff ga = ag\
&\iff a = g^{-1} a g, 
\end{align*}
so 
\begin{align}
ga g^{-1} = a \iff g^{-1} a g = a.
\end{align} 
Therefore
\begin{align*}
C_G (A)&= \{g \in G \mid \forall a \in A,\  g a g^{-1}  =a\}&	ext{ (by definition of $C_G$)}\
&=\{g \in G \mid \forall a \in A,\ g^{-1} a g = a \} &	ext{(by (1))} 
\end{align*}
(Alternatively, since $C_G(A)$ is a subgroup, $g \in C_G(A) \iff g^{-1} \in C_G(A)$.)
\end{proof}

Exercise 2.2.1 ($C_G(A) = \{g \in G \mid g^{-1} a g = a \text{ for all } a \in A\}$)

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Exercise 2.2.1 ($C_G(A) = \{g \in G \mid g^{-1} a g = a \text{ for all } a \in A\}$)

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