Transposition (Mathematics)
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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 ...
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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