cycle notation
The cycle notation is a useful convention for writing down a permutations in terms of its constituent cycles. Let be a finite set, and
distinct elements of . The expression denotes the cycle whose action is
Note there are different expressions for the same cycle; the following all represent the same cycle:
Also note that a 1-element cycle is the same thing as the identity permutation, and thus there is not much point in writing down such things. Rather, it is customary to express the identity permutation simply as or .
Let be a permutation of , and let
be the orbits of with more than 1 element. For each let denote the cardinality of . Also, choose an , and define
We can now express as a product of disjoint cycles, namely
By way of illustration, here are the 24 elements of the symmetric group on expressed using the cycle notation, and grouped according to their conjugacy classes:
Title | cycle notation |
---|---|
Canonical name | CycleNotation |
Date of creation | 2013-03-22 12:33:41 |
Last modified on | 2013-03-22 12:33:41 |
Owner | rmilson (146) |
Last modified by | rmilson (146) |
Numerical id | 6 |
Author | rmilson (146) |
Entry type | Definition |
Classification | msc 20B05 |
Classification | msc 05A05 |
Related topic | Cycle2 |
Related topic | Permutation |
Related topic | OneLineNotationForPermutations |