\section{Conjugacy Classes}

\begin{definition}[Conjugacy Class of $\mathbf{a}$]
	Let $a$ and $b$ be elements of a group $G$. We say that $a$ and $b$ are \textit{conjugate} in $G$ (and call $b$ the \textit{conjugate} of $a$) if $xax^{-1}=b$ for some $x$ in $G$. The \textit{conjugacy class of $a$} is the set $\cl(a) = \{xax^{-1}\ \vert\ x \in G\}$.
\end{definition}

\begin{theorem}[Number of Conjugates of $\mathbf{a}$]
	Let $G$ be a finite group and let $a$ be an element of $G$. Then, $\abs{\cl(a)} = \abs{G:C(a)}$.
\end{theorem}

\begin{corollary}[$\mathbf{\abs{\cl(a)}}$ Divides $\mathbf{\abs{G}}$]
	In a finite group, $\abs{\cl(a)}$ divides $\abs{G}$.
\end{corollary}