# 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 Transposition 2013-03-22 12:24:30 2013-03-22 12:24:30 drini (3) drini (3) 6 drini (3) Definition msc 03-00 msc 05A05 msc 20B99 Cycle2 SignatureOfAPermutation