AKA $k$-cycle. > [!NOTE] Definition 1 (Cyclic Permutation) > Let $\rho\in S_{n}$ be a [[Permutation of Finite Degree|permutation of n letters]]. Then $\rho$ is a *cyclic permutation* of *length* $k$ (or $k$-cycle) iff there exists $k\in\mathbb{N}^{+}$ and $i\in \mathbb{Z}$ such that $k$ is the smallest positive integer such that $\rho^{k}(i)=i$ and $\rho$ fixes each $j$ not in $\{ i, \rho(i),\dots,\rho^{k-1}(i) \}.$ **Notation**: $\rho$ is usually denoted using [[Cycle Notation|cycle notation]] as $(i,\rho (i), \rho^{2}(i),\dots,\rho^{k-1}(i)).$ That is, a $k$-cycle can be represented by a [[List|list]] of length $k$ whose elements are taken from the finite set $\mathbb{N}^{+}_{\leq n}$ without any repeats. However, the $k$-cycle itself does not depend on the starting element of the list. # Properties By [[Number of k-Cycles in nth Symmetric Group]], the number of elements of $S_{n}$ that are a cyclic is given by $\frac{n!}{k(n-k)!}.$ **Product**: Note that [[Disjoint Permutations Commute|disjoint cycles commute]]. **Order**: [[Order of k-Cycle is k]]. **Parity**: Note that [[k-cycles can be factored into 2-cycles]]. Has the same parity as $k-1.$ # Applications **Cycle decomposition**: Note that [[Existence of Disjoint Cycle Decomposition for Permutations of n Letters|every n-permutation can be written uniquely as a product of disjoint cycles, up to the number of factors]].