\section{Free Group}

\begin{theorem}[Equivalence Classes Form a Group]
	Let $S$ be a set of distinct symbols. For any word $u$ in $W(S)$, let $\overline{u}$ denote the set of all words in $W(S)$ equivalent to $u$ (that is, $\overline{u}$ is the equivalence class containing $u$). Then the set of all equivalence classes of elements of $W(S)$ is a group under the operation $\overline{u}\cdot\overline{v} = \overline{uv}$. This group is called a \textit{free group on $S$}.
\end{theorem}

\begin{theorem}[Universal Mapping Property]
	Every group is a homomorphic image of a free group.
\end{theorem}

\begin{corollary}[Universal Factor Group Property]
	Every group is isomorphic to a factor group of a free group.
\end{corollary}
