\section{The Class Equation}

\begin{corollary}[Class Equation]
	For any finite group $G$,
	\[ \abs{G} = \sum \abs{G:C(a)} \]
	where the sum runs over one element of $a$ from each conjugacy class of $G$.
\end{corollary}

\begin{theorem}[$\mathbf{p}$-Groups Have Nontrivial Centers]
	Let $G$ be a nontrivial finite group whose order is a power of a prime $p$. Then $\Z(G)$ has more than one element.
\end{theorem}

\begin{corollary}[Groups of Order $\mathbf{p^2}$ Are Abelian]
	If $\abs{G}=p^2$, where $p$ is prime, then $G$ is Abelian.
\end{corollary}