High Quality Content by WIKIPEDIA articles! 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 with i neq j such that f(ai) = aj, f(aj) = ai and f(ak) = ak for all other indices k. This is often denoted (in the cycle notation) as (ai,aj).Any permutation can be expressed as the composition (product) of transpositions – formally, they are generators for the group. In fact, if one orders the set as in {1,2,3,4,5}, then any permutation can be expressed as a product of adjacent transpositions, meaning the transpositions (k,k + 1), in this case (12),(23),(34),(45). This follows because an arbitrary transposition can be expressed as the product of adjacent transpositions. Concretely, one can express the transposition (k,l) where k < l by moving k to l one step at a time, then moving l back to where k was, which interchanges these two and makes no other changes: (k,l) = (k,k+1)(k+1,k+2)dots(l-1,l)(l-2,l-1)dots(k,k+1).

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