# 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 $g^{\prime}=hgh^{-1}$. Conjugacy of elements is an equivalence relation, and the equivalence classes of $G$ are called .

Two subsets $S$ and $T$ of $G$ are said to be 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.

Title conjugacy class ConjugacyClass 2013-03-22 12:18:09 2013-03-22 12:18:09 djao (24) djao (24) 5 djao (24) Definition msc 20A05 conjugate conjugate set conjugate subgroup ConjugacyClassFormula