# symmetric group

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 .

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 $SL_{n-1}$).

Title symmetric group SymmetricGroup 2013-03-22 12:01:53 2013-03-22 12:01:53 bwebste (988) bwebste (988) 11 bwebste (988) Definition msc 20B30 Group Cycle2 CayleyGraphOfS_3 Symmetry2