In combinatorial mathematics, the cycle notation is a useful convention for writing down a permutation in terms of its constituent cycles. This is also called circular notation and the permutation called a cyclic or circular permutation.
Let be the set , and
be distinct elements of . The expression
denotes the cycle σ whose action is
For each index i,
where is taken to mean .
There are different expressions for the same cycle; the following all represent the same cycle:
Permutation as product of cycles
Let be a permutation of , and let
be the orbits of with more than 1 element. Consider an element , , let denote the cardinality of , =. Also, choose an , and define
We can now express as a product of disjoint cycles, namely
Since disjoint cycles commute with each other, the meaning of this expression is independent of the convention used for the order in products of permutations, namely whether the factors in such a product operate rightmost-first (as is usual more generally for function composition), or leftmost-first as some authors prefer. The meaning of individual cycles is also independent of this convention, namely always as described above.
- Fraleigh 2002:89; Hungerford 1997:230
- Dehn 1930:19
- Hungerford 1997:231
- Johnson 2003:691
- Dehn, Edgar (1960) , Algebraic Equations, Dover.
- Fraleigh, John (2003), A first course in abstract algebra (7th ed.), Addison Wesley, p. 88–90, ISBN 978-0-201-76390-4.
- Hungerford, Thomas W. (1997), Abstract Algebra: An Introduction, Brooks/Cole, ISBN 978-0-03-010559-3.
- Johnson, James L. (2003), Probability and Statistics for Computer Science, Wiley Interscience, ISBN 978-0-471-32672-4.