Given a finite setMathworldPlanetmath X={a1,a2,,an}, a transpositionMathworldPlanetmath is a permutationMathworldPlanetmath (bijective function of X onto itself) f such that there exist indices i,j such that f(ai)=aj, f(aj)=ai and f(ak)=ak for all other indices k. This is often denoted (in the cycle notation) as (a,b).

Example: If X={a,b,c,d,e} the function σ given by

σ(a) = a
σ(b) = e
σ(c) = c
σ(d) = d
σ(e) = b

is a transposition.

One of the main results on symmetric groupsMathworldPlanetmathPlanetmath states that any permutation can be expressed as compositionMathworldPlanetmath (productPlanetmathPlanetmath) of transpositions, and for any two decompositions of a given permutation, the number of transpositions is always even or always odd.

Title transposition
Canonical name Transposition
Date of creation 2013-03-22 12:24:30
Last modified on 2013-03-22 12:24:30
Owner drini (3)
Last modified by drini (3)
Numerical id 6
Author drini (3)
Entry type Definition
Classification msc 03-00
Classification msc 05A05
Classification msc 20B99
Related topic Cycle2
Related topic SignatureOfAPermutation