Definition:Symmetric Group

Definition
Let $S$ be a set.

Let $\Gamma \paren S$ be the set of permutations on $S$.

Let $\struct {\Gamma \paren S, \circ}$ be the algebraic structure such that $\circ$ denotes the composition of mappings.

Then $\struct {\Gamma \paren S, \circ}$ is called the symmetric group on $S$.

If $S$ has $n$ elements, then $\struct {\Gamma \paren S, \circ}$ is often denoted $S_n$.

Also see

 * Symmetric Group is Group


 * Symmetric Groups of Same Order are Isomorphic

If $S$ is finite with cardinality $n$, then:
 * Order of Symmetric Group: the order of $S_n$ is $n!$