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For , with denoting the symmetric group on elements, consider a permutation of elements.
Write
for the cyclic group generated by . This cyclic group (whose order is the order of ) canonically acts on the set through the defining group action of on this set.
Then a cycle of the permutation is an orbit of with respect to this canonical group action on .
More explicitly, this means that a cycle of is a subset of the form
for any .
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 17:33:10. See the history of this page for a list of all contributions to it.