\begin{align*}
Z(G)=\{g\in G: gx=xg \text{ for every } x\in G\}.
\end{align*}
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Let $\operatorname{Aut}(G)$ denote the group of automorphisms of $G$, and let $\operatorname{Inn}(G)\le \operatorname{Aut}(G)$ be the subgroup consisting of automorphisms of the form $x\mapsto gxg^{-1}$ for some $g\in G$. Then