\begin{definition}[Definition:Symmetric Group]
Let $S$ be a [[Definition:Set|set]].
Let $\map \Gamma S$ denote the [[Definition:Set|set]] of [[Definition:Permutation|permutations]] on $S$.
Let $\struct {\map \Gamma S, \circ}$ be the [[Definition:Algebraic Structure|algebraic structure]] such that $\circ$ denotes the [[Definition:Composition of Mappings|composition of mappings]].
Then $\struct {\map \Gamma S, \circ}$ is called the '''symmetric group on $S$'''.
If $S$ has $n$ [[Definition:Element|elements]], then $\struct {\map \Gamma S, \circ}$ is often denoted $S_n$.
\end{definition}
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