Let $X$ be a set. Let ${\rm Sym}(X)$ be the set of permutations of $X$ (i.e. the set of bijective functions from $X$ to itself). Then the act of taking the composition of two permutations induces a group structure on ${\rm Sym}(X)$ We call this group the symmetric group.

The group ${\rm Sym}(\{1,2,\ldots, n\})$ is often denoted $S_n$ or $\mathfrak{S}_n$

$S_n$ is generated by the transpositions $\{(1,2),(2,3),\ldots,(n-1,n)\}$ and by any pair of a 2-cycle and $n$ cycle.

$S_n$ is the Weyl group of the $A_{n-1}$ root system (and hence of the special linear group $SL_{n-1}$ .