20A05-ConjugacyClass.tex

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\pmtitle{conjugacy class}
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\pmtype{Definition}
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\pmsynonym{conjugate}{ConjugacyClass}
\pmsynonym{conjugate set}{ConjugacyClass}
\pmsynonym{conjugate subgroup}{ConjugacyClass}
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Two elements $g$ and $g'$ of a group $G$ are said to be {\em conjugate} if there exists $h \in G$ such that $g' = hgh^{-1}$. Conjugacy of elements is an equivalence relation, and the equivalence classes of $G$ are called {\em conjugacy classes}.

Two subsets $S$ and $T$ of $G$ are said to be {\em conjugate} if there exists $g \in G$ such that
$$
T = \{gsg^{-1} \mid s \in S\} \subset G.
$$
In this situation, it is common to write $gSg^{-1}$ for $T$ to denote the fact that everything in $T$ has the form $gsg^{-1}$ for some $s \in S$. We say that two subgroups of $G$ are conjugate if they are conjugate as subsets.
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