The symmetric group on a set $X$ is the group
\[\SS_X = \bij(X, X)\]equipped with composition. Each element of $\SS_X$ is called a permutation of $X$.
For any $n \in \N^+$, the symmetric group of degree $n$ is
\[\SS_n = \SS_{\llbra n \rrbra}.\]
Proof. ($\SS_X$ is a group) By COMP > Proposition 5, $\fun(X, X)$ is a monoid. Since
\[\bij(X, X) = \inv\fun(X, X),\]$\bij(X, X)$ is a group by GRP > Proposition 6.