Category:Definitions/Cyclic Permutations

This category contains definitions related to Cyclic Permutations.
Related results can be found in Category:Cyclic Permutations.

Let $S_n$ denote the symmetric group on $n$ letters.

Let $\rho \in S_n$ be a permutation on $S$.

Then $\rho$ is a cyclic permutation of length $k$ if and only if there exists $k \in \Z: k > 0$ and $i \in \Z$ such that:

$(1): \quad k$ is the smallest such that $\map {\rho^k} i = i$

$(2): \quad \rho$ fixes each $j$ not in $\set {i, \map \rho i, \ldots, \map {\rho^{k - 1} } i}$.