Editing Cyclic permutation (section)

{{short description|Type of (mathematical) permutation with no fixed element}}
{{other uses|Cyclic (mathematics)}}
In [[mathematics]], and in particular in [[group theory]], a '''cyclic permutation''' is a [[permutation]] consisting of a single cycle.<ref name=":0">{{Cite book |last=Gross |first=Jonathan L. |title=Combinatorial methods with computer applications |date=2008 |publisher=Chapman & Hall/CRC |isbn=978-1-58488-743-0 |series=Discrete mathematics and its applications |location=Boca Raton, Fla. |pages=29}}</ref><ref name=":1">{{Cite book |last=Knuth |first=Donald E. |title=The Art of Computer Programming |publisher=Addison-Wesley |year=2002 |pages=35}}</ref> In some cases, cyclic permutations are referred to as '''cycles''';<ref name=":2">{{Cite book |last=Bogart |first=Kenneth P. |title=Introductory combinatorics |date=2000 |publisher=Harcourt Academic Press |isbn=978-0-12-110830-4 |edition=3 |location=London |pages=554}}</ref> if a cyclic permutation has ''k'' elements, it may be called a '''''k''-cycle'''. Some authors widen this definition to include permutations with fixed points in addition to at most one non-trivial cycle.<ref name=":2" /><ref name=":3">{{Cite book |last=Rosen |first=Kenneth H. |title=Handbook of discrete and combinatorial mathematics |date=2000 |publisher=CRC press |isbn=978-0-8493-0149-0 |location=Boca Raton London New York}}</ref> In [[Permutation#Cycle notation|cycle notation]], cyclic permutations are denoted by the list of their elements enclosed with parentheses, in the order to which they are permuted.

For example, the permutation (1 3 2 4) that sends 1 to 3, 3 to 2, 2 to 4 and 4 to 1 is a 4-cycle, and the permutation (1 3 2)(4) that sends 1 to 3, 3 to 2, 2 to 1 and 4 to 4 is considered a 3-cycle by some authors. On the other hand, the permutation (1 3)(2 4) that sends 1 to 3, 3 to 1, 2 to 4 and 4 to 2 is not a cyclic permutation because it separately permutes the pairs {1, 3} and {2, 4}.

For the wider definition of a cyclic permutation, allowing fixed points, these fixed points each constitute trivial [[orbit (group theory)|orbit]]s of the permutation, and there is a single non-trivial orbit containing all the remaining points. This can be used as a definition: a cyclic permutation (allowing fixed points) is a permutation that has a single non-trivial orbit. Every permutation on finitely many elements can be decomposed into cyclic permutations whose non-trivial orbits are disjoint.<ref>{{cite book|title=Fundamental Concepts of Abstract Algebra|series=Dover Books on Mathematics|first=Gertrude|last=Ehrlich|publisher=Courier Corporation|year=2013|isbn=9780486291864|page=69|url=https://books.google.com/books?id=1dGjAQAAQBAJ&pg=PA69}}</ref>

The individual cyclic parts of a permutation are also called '''[[Cycles and fixed points|cycles]]''', thus the second example is composed of a 3-cycle and a 1-cycle (or ''fixed point'') and the third is composed of two 2-cycles.