transposition

Given a finite set $X=\{a_{1},a_{2},\ldots,a_{n}\}$ , a transposition is a permutation (bijective function of $X$ onto itself) $f$ such that there exist indices $i, j$ such that $f(a_{i})=a_{j}$ , $f(a_{j})=a_{i}$ and $f(a_{k})=a_{k}$ 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 $\sigma$ given by

$\displaystyle\sigma(a)$	$\displaystyle=$	$\displaystyle a$
$\displaystyle\sigma(b)$	$\displaystyle=$	$\displaystyle e$
$\displaystyle\sigma(c)$	$\displaystyle=$	$\displaystyle c$
$\displaystyle\sigma(d)$	$\displaystyle=$	$\displaystyle d$
$\displaystyle\sigma(e)$	$\displaystyle=$	$\displaystyle b$

is a transposition.

One of the main results on symmetric groups states that any permutation can be expressed as composition (product) 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