Exercise 2.2.3 ($A \subseteq B \Rightarrow C_G(B) \leq C_G(A)$)

Question

Prove that if $A$ and $B$ are subsets of $G$ with $A \subseteq B$ then $C_G(B)$ is a subgroup of $C_G(A)$.

Richard Ganaye · Accepted Answer

\begin{proof} 
Let $a \in C_G(B)$. Then for all $ x \in B$, $ax = xa$.

A fortiori, since $A \subseteq B$, for all $x \in A$, $ax = xa.$

Therefore $x \in C_G(A)$. This shows that $C_G(B) \subseteq C_G(A).$

$$A \subseteq B \Rightarrow C_G(B) \leq  C_G(A).$$

\end{proof}

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