Let $G$ be a group with operation written multiplicatively, and let $A \subset G$. For each $g \in G$, define the [conjugacy class](/page/Conjugacy%20Class) of $g$ in $G$ by
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\begin{align*}
\operatorname{Cl}_G(g) := \{hgh^{-1} : h \in G\}.
\end{align*}
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Then $A$ is conjugation-invariant, meaning that
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\begin{align*}
\forall a \in A,\ \forall h \in G,\quad hah^{-1} \in A,
\end{align*}
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if and only if $A$ is a union of conjugacy classes of $G$, meaning that there exists a subset $S \subset G$ such that
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\begin{align*}
A = \bigcup_{s \in S} \operatorname{Cl}_G(s).
\end{align*}