Created the Abstract Algebra theorems and definitions cheat sheet

part-5/chapters/chapter-24/conjugacy-classes.tex

@@ -0,0 +1,13 @@
+\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}