Exercise 2.2.11 (Center and normalizer of a subset $A$ of $G$)

Proof. We know that $C_G(A)\subseteq N_G(A)$ (see p.50). Moreover $Z(G)\subseteq C_G(A)$ (If $g\in G$ commutes with every element of $G$, it commutes in particular with the elements of $A\subseteq G$). So

This shows that

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