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.

How Cycle Notation Works

Here are some examples of cycles:

(1 3). This function sends 1 to 3 and 3 to 1. In algebraic notation, f(1) = 3 and f(3) = 1. All of the other numbers (0, 2, and 4 through 9) are fixed by this function.

(4 8 3). This function sends 4 to 8, 8 to 3, and 3 to 4. All of the other numbers are fixed by this function.

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.

The Applet

In the applet below, students can practice translating from familiar function notation to cycle notation.

Use the drop-boxes on the right to create your function. Make sure that the function is one-to-one and onto. Only one-to-one correspondences are considered permutations!

Alternatively, click the Randomize button to have the program generate a random permutation for you.

Stop! On your own, figure out the cycle notation that corresponds to the function.

Click the Generate button to generate the cycle notation and check that your answer is correct.

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: