Transposition

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 $X$ is a permutation on $X$ that swaps two elements of $X$ 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.

Transpositions generate the finitary symmetric group, that is, given a finite set $X$, the set of all transpositions in $Sym(X)$ will generate $Sym(X)$.