For $n \in \mathbb{N}$, with $Sym(n)$ denoting the symmetric group on $n$ elements, consider a permutation $\sigma \;\in\; Sym(n)$ of $n$ elements.
Write
for the cyclic group generated by $\sigma$. This cyclic group (whose order is the order of $\sigma$) canonically acts on the set $\{1, \cdots, n\}$ through the defining group action of $Sym(n)$ on this set.
Then a cycle of the permutation $\sigma$ is an orbit of $\langle \sigma \rangle$ with respect to this canonical group action on $\{1, \cdots, n\}$.
More explicitly, this means that a cycle of $\sigma$ is a subset of the form
for any $1 \leq k \leq n$.
Beware that this is really meant as a subset, not as a tuple. So for instance
is the same cycle.
The permutation action restricted to the cycle is a cyclic permutation.
See also
Wikipedia, Cyclic permutation
MathWorld, Permutation Cycle
Last revised on April 18, 2021 at 13:33:10. See the history of this page for a list of all contributions to it.