AKA symmetric group of degree $n$; symmetric group of $n$ letters/objects. > [!NOTE] Definition ($S_{n}$) > Let $n\in\mathbb{N}^{+}.$ The $n$th symmetric group, denoted $S_{n},$ is the [[Symmetric Group|symmetric group]] of $\mathbb{N}^{+}_{\leq n} = \{ k\in \mathbb{N}^{+} \mid k \leq n \}=\{ 1,2,\dots,n \}$which is the set of [[Permutation of Finite Degree|permutations of n letters]] under [[Function Composition|function compositon]]. **Notation**: It is convenient to refer to the elements of $S_{n}$ using [[Two-Row Notation|two-row notation]] or [[Cycle Notation|cycle notation]]. Also we use [[Multiplicative Notation|multiplicative notation]] for function composition thus $\pi\circ\rho$ is written $\pi \rho$ for all $\pi,\rho\in S_{n}.$ > [!Example] > $S_{3}$ has $6$ elements: $\begin{aligned}&\begin{bmatrix}1&2&3\\1&2&3\end{bmatrix}\quad\begin{bmatrix}1&2&3\\1&3&2\end{bmatrix}\quad\begin{bmatrix}1&2&3\\2&1&3\end{bmatrix}\\\\&\begin{bmatrix}1&2&3\\2&3&1\end{bmatrix}\quad\begin{bmatrix}1&2&3\\3&1&2\end{bmatrix}\quad\begin{bmatrix}1&2&3\\3&2&1\end{bmatrix}\end{aligned}$expressed in two-row notation. # Properties By [[Number of Permutations of n Letters]], $|S_{n}|=n!.$ **Isomorphisms**: Suppose that $A$ is a [[Finite Set|finite set]] with cardinality $n,$ then $S_{n}$ and $\text{Sym}(A)$ are [[Symmetric Groups of Same Order Are Isomorphic|isomorphic]]. **Subgroups**: [[Parity of a Permutation of n letters|Even permutations]] of $n$ letters form a group known as the [[nth Alternating Group|alternating group]], denoted $A_{n}.$ # Applications **Determinant of square matrix**: ... [[Determinant]].