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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 onto itself) such that there exist indices such that , and for all other indices This is often denoted (in the cycle notation) as Example: If the function given by is a transposition. One of the main results on symmetric groups states that any permutation can be expressed as the composition (product) of transpositions, and for any decomposition of a given permutation into transpositions, the number of transpositions is always even or always odd.
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