Cyclic permutation

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, and in particular in <a href="/facts/Group_theory/NfowcI5H">group theory</a>, a cyclic permutation is a <a href="/facts/Permutation/JSw9ukmv">permutation</a> 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 <a href="/facts/Permutation/JSw9ukmv">cycle notation</a>, 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 each constitute trivial <a href="/facts/Orbit_(group_theory)/Vh9VtUUw">orbits</a> of the permutation, and there is a single non-trivial orbit containing all the remaining points. This can be used as a definition: a cyclic permutation (allowing fixed points) is a permutation that has a single non-trivial orbit. Every permutation on finitely many elements can be decomposed into cyclic permutations whose non-trivial orbits are disjoint.
The individual cyclic parts of a permutation are also called <a href="/facts/Cycles_and_fixed_points/1zjqh9he">cycles</a>, thus the second example is composed of a 3-cycle and a 1-cycle (or fixed point) and the third is composed of two 2-cycles.

Cyclic permutation open-in-new

Cyclic permutation