Transposition (mathematics)

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In informal language, a transposition is a function that swaps two elements of a set. More formally, given a finite set , 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).

For example, if X = {a,b,c,d,e}, the function σ given by

is a transposition.

Any permutation can be expressed as the composition (product) of transpositions. One of the main results on symmetric groups states that either all of the decompositions of a given permutation into transpositions have an even number of transpositions, or they all have an odd number of transpositions, that allows to define the parity of a permutation.

[edit] External links

• Permutations as a Product of Transpositions

[edit] See also

• Cycle (mathematics)
• Signature of a permutation

This article incorporates material from transposition on PlanetMath, which is licensed under the GFDL.