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Hometransposition

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# 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.

Related:

Cycle2, SignatureOfAPermutation

Type of Math Object:

Definition

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## Mathematics Subject Classification

03-00*no label found*05A05

*no label found*20B99

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