Transposition: Difference between revisions
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A '''transposition''' on a set <math>X</math> is a [[permutation]] on <math>X</math> that swaps two elements of <math>X</math> and fixes the rest. | A '''transposition''' on a set <math>X</math> is a [[permutation]] on <math>X</math> that swaps two elements of <math>X</math> and fixes the rest. | ||
==Facts== | |||
Any permutation can be written as the product of some transpositions. The number of transpositions that a permutation can be written as is not well-defined, the permutation can be written as a product of transpositions in different ways, of different lengths. However, the parity of the length is well-defined (i.e. whether it is written as the product of an even number or an odd number of transpositions.) The sign of a permutation is defined to be 1 if the permutation can be written as the product of an even number of permutations, and -1 otherwise. | |||
Revision as of 15:50, 14 December 2023
This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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Definition
A transposition on a set is a permutation on that swaps two elements of and fixes the rest.
Facts
Any permutation can be written as the product of some transpositions. The number of transpositions that a permutation can be written as is not well-defined, the permutation can be written as a product of transpositions in different ways, of different lengths. However, the parity of the length is well-defined (i.e. whether it is written as the product of an even number or an odd number of transpositions.) The sign of a permutation is defined to be 1 if the permutation can be written as the product of an even number of permutations, and -1 otherwise.