![]() This permits the parity of a permutation to be a well-defined concept. ![]() 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. In fact, the symmetric group is a Coxeter group, meaning that it is generated by elements of order 2 (the adjacent transpositions), and all relations are of a certain form. On the other hand, the permutation (1 3)(2 4) that sends 1 to 3, 3 to 1, 2 to 4 and 4 to 2 is not a cyclic permutation because it separately permutes the pairs In cycle notation, cyclic permutations are denoted by the list of their elements enclosed with parentheses, in the order to which they are permuted.įor example, the permutation (1 3 2 4) that sends 1 to 3, 3 to 2, 2 to 4 and 4 to 1 is a 4-cycle, and the permutation (1 3 2)(4) that sends 1 to 3, 3 to 2, 2 to 1 and 4 to 4 is considered a 3-cycle by some authors. Some authors widen this definition to include permutations with fixed points in addition to at most one non-trivial cycle. In some cases, cyclic permutations are referred to as cycles if a cyclic permutation has k elements, it may be called a k-cycle. ![]() In mathematics, and in particular in group theory, a cyclic permutation is a permutation consisting of a single cycle. For other uses, see Cyclic (mathematics). ![]()
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