conjugacy class

Two elements $g$ and $g^{\prime }$ of a group $G$ are said to be conjugate if there exists $h\in G$ such that $g^{\prime }=hg{h}^{-1}$. Conjugacy of elements is an equivalence relation, and the equivalence classes of $G$ are called conjugacy classes.

Two subsets $S$ and $T$ of $G$ are said to be conjugate if there exists $g\in G$ such that

$$T=\{gs{g}^{-1}\mid s\in S\}\subset G.$$

In this situation, it is common to write $gS{g}^{-1}$ for $T$ to denote the fact that everything in $T$ has the form $gs{g}^{-1}$ for some $s\in S$. We say that two subgroups of $G$ are conjugate if they are conjugate as subsets.