TRANSPOSITION (MATHEMATICS)

In informal language, a 'transposition' is a function that swaps two elements of a set. More formally, 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
:
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.

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★ Permutations as a Product of Transpositions

See also

★ cycle

★ signature of a permutation