cycle notation

The cycle notation is a useful convention for writing down a permutationsMathworldPlanetmath in terms of its constituent cycles. Let S be a finite setMathworldPlanetmath, and


distinct elements of S. The expression (a1,,ak) denotes the cycle whose action is


Note there are k 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 identityPlanetmathPlanetmathPlanetmathPlanetmath 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 (1).

Let π be a permutation of S, and let


be the orbits of π with more than 1 element. For each j=1,,k let nj denote the cardinality of Sj. Also, choose an a1,jSj, and define


We can now express π as a productPlanetmathPlanetmathPlanetmath of disjoint cycles, namely


By way of illustration, here are the 24 elements of the symmetric groupMathworldPlanetmathPlanetmath on {1,2,3,4} expressed using the cycle notation, and grouped according to their conjugacy classesMathworldPlanetmathPlanetmath:

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