nLab
cycle of a permutation

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Definition

For nn \in \mathbb{N}, with Sym(n)Sym(n) denoting the symmetric group on nn elements, consider a permutation σSym(n)\sigma \;\in\; Sym(n) of nn elements.

Write

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

for the cyclic group generated by σ\sigma. This cyclic group (whose order is the order of σ\sigma) canonically acts on the set {1,,n}\{1, \cdots, n\} through the defining group action of Sym(n)Sym(n) on this set.

Then a cycle of the permutation σ\sigma is an orbit of σ\langle \sigma \rangle with respect to this canonical group action on {1,,n}\{1, \cdots, n\}.

More explicitly, this means that a cycle of σ\sigma is a subset of the form

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

for any 1kn1 \leq k \leq n.

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

{σ(k),σ(σ(k)),σ 3(k)),}{1,,n} \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.

References

See also

Last revised on April 18, 2021 at 13:33:10. See the history of this page for a list of all contributions to it.