- Cycle notation
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For the cyclic decomposition of graphs, see Cycle decomposition (graph theory). For cycling terminology, see glossary of bicycling.
In combinatorial mathematics, the cycle notation is a useful convention for writing down a permutation in terms of its constituent cycles.[1] This is also called circular notation and the permutation called a cyclic or circular permutation.[2]
Contents
Definition
be distinct elements of S. The expression
denotes the cycle σ whose action is
For each index i,
- σ(ai) = ai + 1,
where ak + 1 is taken to mean a1.
There are k different expressions for the same cycle; the following all represent the same cycle:
A 1-element cycle such as (3) is the identity permutation.[3] The identity permutation can also be written as an empty cycle, "()".[4]
Permutation as product of cycles
Let π be a permutation of S, and let
be the orbits of π with more than 1 element. Consider an element Sj,
, let nj denote the cardinality of Sj, | Sj | =nj. Also, choose an
, and define
We can now express π as a product of disjoint cycles, namely
Note that the usual convention in cycle notation is to multiply from left to right (in contrast with composition of functions, which is normally done from right to left). For example, the product
is equal to
not
.
Example
Here are the 24 elements of the symmetric group on {1,2,3,4} expressed using the cycle notation, and grouped according to their conjugacy classes:
See also
Notes
References
- Dehn, Edgar (1960) [1930], Algebraic Equations, Dover.
- Fraleigh, John (2003), A first course in abstract algebra (7th ed.), Addison Wesley, p. 88–90, ISBN 978-0201763904.
- Hungerford, Thomas W. (1997), Abstract Algebra: An Introduction, Brooks/Cole, ISBN 978-0030105593.
- Johnson, James L. (2003), Probability and Statistics for Computer Science, Wiley Interscience, ISBN 978-0471326724.
This article incorporates material from cycle notation on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
Categories:- Permutations
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