Symmetric Group

Definition 1. 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 G > Proposition 2.