AKA alternating group of $n$ letters; $n$th alternating group. > [!NOTE] Definition ($A_{n}$) > Let $n\in \mathbb{N}^{+}.$ The $n$th alternating group is the set of [[Parity of a Permutation of n letters|even]] [[Permutation of Finite Degree|permutation of n letters]], denoted $A_{n}= \{ \sigma\in S_{n} \mid \text{sgn} (\sigma) = 1\}$where $S_{n}$ is the [[Symmetric Groups of Finite Degree|nth symmetric group]]. > # Properties **Algebra**: Note that $A_{n}$ is a [[Alternating Group is a Subgroup of Symmetric Group|subgroup]] of $S_{n}.$ Moreover, it is a [[Alternating Group is a Normal Subgroup of Symmetric Group|normal subgroup]] of $A_{n}.$ **Size**: $A_{n}$ has [[Order of nth Alternating Group|order]] $n!/2.$