symmetric group

Let X be a set. Let Sym(X) be the set of permutationsMathworldPlanetmath of X (i.e. the set of bijective functions from X to itself). Then the act of taking the compositionMathworldPlanetmathPlanetmath of two permutations induces a group structureMathworldPlanetmath on Sym(X). We call this group the symmetric groupMathworldPlanetmathPlanetmath.

The group Sym({1,2,,n}) is often denoted Sn or 𝔖n.

Sn is generated by the transpositionsMathworldPlanetmath {(1,2),(2,3),,(n-1,n)}, and by any pair of a 2-cycle and n-cycle.

Sn is the Weyl group of the An-1 root system (and hence of the special linear groupMathworldPlanetmath SLn-1).

Title symmetric group
Canonical name SymmetricGroup
Date of creation 2013-03-22 12:01:53
Last modified on 2013-03-22 12:01:53
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 11
Author bwebste (988)
Entry type Definition
Classification msc 20B30
Related topic Group
Related topic Cycle2
Related topic CayleyGraphOfS_3
Related topic Symmetry2