\section{Elementary Properties of Groups}

\begin{theorem}[Uniqueness of the Identity]
	In a group $G$, there is only one identity element.
\end{theorem}

\begin{theorem}[Cancellation]
	In a group $G$, the right and left cancellation laws hold; that is, $ba = ca$ implies $b = c$, and $ab = ac$ implies $b = c$.
\end{theorem}

\begin{theorem}[Uniqueness of Inverses]
	For each element $a$ in a group $G$, there is a unique element $b$ in $G$ such that $ab = ba = e$.
\end{theorem}

\begin{theorem}[Socks-Shoes Property]
	For group elements $a$ and $b$, $(ab)^{-1} = b^{-1}a^{-1}$.
\end{theorem}