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Centers and conjugacy classes of groups relative to a pair of group homomorphisms

$\newcommand{\defeq}{\mathbin{\overset{\mathrm{def}}{=}}}$Given a group $G$, its center $\mathrm{Z}(G)$ and set of conjugacy classes $\mathrm{Cl}(G)$ are defined by \begin{align*} \mathrm{Z}(G) &\...