The symmetric group on n letters, Sym(n), is one of the first examples of non-abelian groups that students learn in an undergraduate group theory course. One of the challenges students face is understanding the cycle notation. Students are familiar with "bubble and arrow" diagrams to represent functions. However, since the functions contained in Sym(n) are necessarily one-to-one and onto, cycle notation provides a convenient shorthand to represent these objects.
While the n objects being permuted could be anything (letters, numbers, etc.), we'll use numbers from 0 to n-1. To keep things relatively small, in the applet below we use n = 10. Of course, this means we still have 10! possible permutations to consider.
Here are some examples of cycles:
One of the basic theorems relating to symmetric groups states that each permutation can be written as the composition of disjoint cycles. Here "disjoint" means that the cycles do not permute the same numbers. So (1 6)(4 3 9 5) is a composition of disjoint cycles, but (5 2)(7 2 9) is not.
In the applet below, students can practice translating from familiar function notation to cycle notation.
Note that cycle notation can be written in different ways. This program will always write the cycles so that the smallest number in the cycle is written first, and the cycles are sorted in order of these first numbers.
The cycle decomposition (1 5)(2 4 3) can be written in many different ways:
This page was created by James Hamblin, Associate Professor of Mathematics at Shippensburg University.