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]
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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:
A 1-element cycle such as (3) is the identity permutation.[3] The identity permutation can also be written as an empty cycle, "()".[4]
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
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 .
Here are the 24 elements of the symmetric group on expressed using the cycle notation, and grouped according to their conjugacy classes:
This article incorporates material from cycle notation on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.