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.