# transposition

Given a finite set^{} $X=\{{a}_{1},{a}_{2},\mathrm{\dots},{a}_{n}\}$, 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

$\sigma (a)$ | $=$ | $a$ | ||

$\sigma (b)$ | $=$ | $e$ | ||

$\sigma (c)$ | $=$ | $c$ | ||

$\sigma (d)$ | $=$ | $d$ | ||

$\sigma (e)$ | $=$ | $b$ |

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 |