# How to Make a Cayley Table

Instructor: Carmen Andert

Carmen has two master's degrees in mathematics has has taught mathematics classes at the college level for the past 9 years.

In this lesson, we will define and give examples of Cayley Tables. We will also give a process for constructing a Cayley Table and work through an example.

## Cayley Tables

When you studied multiplication in elementary school, you likely had to memorize multiplication tables. These tables had rows and columns of numbers as headings and products of those numbers in the interior of the table. Multiplication tables contain all the relationships between the numbers (at least as long as you only care about multiplication.)

A group is a set of elements closed under an associative operation that includes an identity and an inverse for each element. A Cayley Table describes the operation (i.e., the interaction between the elements) of a finite group. A Cayley Table is really the multiplication table for the group, except that the group operation may not necessarily be multiplication.

#### Example

Let's build the Cayley Table for the cyclic group of order 3 (that just means that it has three elements in the group). Although different symbols can by used, we will call the group elements 0, 1, and 2, and we will use the group operation of addition modulo 3. This means that we add the elements, but if we get a number higher than 2, we subtract 3 automatically. For example, 1+2=0 in this group since 1+2 gives 3, but then we subtract 3 to get 0.

Here is the Cayley Table for this group.

To fill in the bottom row of the Cayley Table, added modulo 3 to get that 2+0=2, 2+1=0, and 2+2=1. Filling in a Cayley Table is easy as long as you know the group operation!

## Properties of a Cayley Table

The Cayley Table gives all the information needed to understand the structure of a group. From the Cayley Table for the group above, we see what the elements are (0, 1, and 2). Did you notice that each row and each column contain each element? Like a correct Sudoku puzzle, this pattern is true for any correct Cayley Table. There are a couple of other things we can figure out about the group from its Cayley Table:

1. Because the 0 row is the same as the heading row and the 0 column is the same as the heading column, we know that 0 is the identity element of the group.
2. How do we find the inverse of a given element? Suppose we want to know the inverse of the element 1 in the cyclic group of order 3. Look in the 1 row and find the identity element 0. Then go up to find the heading of that column--this is the 2 column. Therefore, the inverse of 1 is 2.

## Constructing a Cayley Table

To make a Cayley Table for a given finite group, begin by listing the group elements along the top row and along the left column. Traditionally, the identity element is listed first and the elements are listed in the same order left to right and top to bottom. This part is super easy!

Now all that remains is to fill in the interior of the table. To fill in the element in the row for element x and the column for element y, figure out what x*y is by thinking about the operation for that particular group. Do this for every pair of elements. Notice that x*y and y*x may not always be the same, so be careful about the rows and columns. (Here, the symbol * is representing the group operation, which may be addition, multiplication, composition, or something else depending on what the group is.)

Once you are done constructing a Cayley Table, how do you know if you have done it correctly or not? The only way to be truly sure is to double check each and every entry, but there are a few easy things to look for:

1. Every entry must be an element of the group. You can't 'make up' new elements as you go.
2. The row corresponding to the identity element (usually the top row) must match the heading row.
3. The column corresponding to the identity element (usually the left column) must match the heading column.
4. Each row must contain each group element exactly once. (Like Sudoku)
5. Each column must contain each group element exactly once. (Like Sudoku)

#### Example

You are asked to construct a Cayley Table for the group of symmetries of an equilateral triangle. A symmetry is a rigid motion that leaves the triangle looking the same as its initial position. It may help for you to cut out an equilateral triangle from a piece of paper to maneuver as you think about this. Label the corners of your triangle with the letters A, B, and C as shown and then label the corners on the back of the triangle so that they match the front.

One way to leave the triangle looking the same is to not move it. We will let e be the group element meaning 'do nothing.' We can also rotate the triangle 120° counterclockwise or we can rotate it 240° counterclockwise. Let's call these two maneuvers R₁ and R₂, respectively. Notice that although the labels on the corners of the triangles have shifted, the unlabeled triangle looks the same as it did when you started. I know this seems confusing, but you will get the hang of it by the time we finish!

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