The symmetric group on $X$ is the group
\[\SS_X = \bij(X,X)\]under composition. Each element of $\SS_X$ is called a permutation of $X$.
If $n\in\N^+$, the symmetric group of degree $n$ is
\[\SS_n = \SS_{\llbra n\rrbra}.\]
$\SS_X$ is indeed a group:
Proof.By PROP-CP-FMO, $\fun(X,X)$ is a monoid. Since
\[\bij(X,X) = \inv\fun(X,X),\]$\bij(X,X)$ is a group by PROP-GRP-INV.