Cyclic permutation

Common card · No. 387044

In mathematics, and in particular in group theory, a cyclic permutation is a permutation consisting of a single 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. Some authors widen this definition to include permutations with fixed points in addition to at most one non-trivial cycle. In cycle notation, cyclic permutations are denoted by the list of their elements enclosed with parentheses, in the order to which they are permuted. For 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. 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 {1, 3} and {2, 4}. For the wider definition of a cyclic permutation, allowing fixed points, these fixed points...

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