\begin{definition}[Definition:Alternating Group]
Let $S_n$ denote the symmetric group on $n$ letters.
For any $\pi \in S_n$, let $\map \sgn \pi$ be the sign of $\pi$.
The kernel of the mapping $\sgn: S_n \to C_2$ is called the '''alternating group on $n$ letters''' and denoted $A_n$.
\end{definition}