nLab cycle of a permutation

Definition

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

$$\langle \sigma \rangle \;\coloneqq\; \big\{ e, \sigma, \sigma^2, \cdots , \sigma^{order(\sigma)-1} \big\} \;\subset\; Sym(n)$$

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

$$\big\{ k, \sigma(k), \sigma(\sigma(k)), \cdots \big\} \;\subset\; \{1,\cdots, n\}$$

for any $1 \leq k \leq n$.

Beware that this is really meant as a subset, not as a tuple. So for instance

$$\big\{ \sigma(k), \sigma(\sigma(k)), \sigma^3(k)), \cdots \big\} \;\subset\; \{1,\cdots, n\}$$

is the same cycle.

The permutation action restricted to the cycle is a cyclic permutation.

Related concepts

• Cayley distance, Kendall distance

• cyclic permutation

• orbit

References

See also

• Wikipedia, Cyclic permutation

• MathWorld, Permutation Cycle