EXAMPLE 5 A Group of Non-Numbers

With all your experience with arithmetic, numbers are concrete to you, even if thinking of them in terms of a group isn’t yet familiar. Here, we look at a very simple “abstract” group. The group is a collection of just three elements {A, B, C}, and it is convenient to show how the operation behaves by giving a table of its results as follows:

The table is organized so that, for example, we find the result of A ▴ B by looking in the row for A and the column for B, finding B. So . Similarly, . We confirm that indeed this set is a group under the operation.

Since all the entries in the table are from {A, B, C}, the set is closed under the operation. You should identify which element serves as an identity element. What is the inverse to A? To B? To C? To check associativity would require checking the results of all possible products , where each of X, Y, and Z can be any of A, B, or C. We won’t go to that (tedious) length, but you should check just one example. For instance, , while . Also, can you see why there are products to check?

This particular abstract group can be interpreted concretely in several ways. One interpretation is in terms of an equilateral triangle.

Each of A, B, and C is a rotation of the triangle about its center. A is a rotation by 0°, B is a rotation clockwise by 120° (one-third of a complete turn), and C is a rotation counterclockwise by 120°. The operation ▴ just gives the result of doing one rotation followed by another. For example, is first to rotate the triangle 120° clockwise, then rotate it 120° counterclockwise—which leaves it as if it had not rotated at all; that is, as rotated by 0°. Hence, .

We have, in fact, explored here some of the symmetries of an equilateral triangle. (What other symmetries does an equilateral triangle have? Think of it as a rosette pattern.)