We have seen that a specific group can be defined in these ways:

(i) A set of elements with a binary operation satisfying the three group axioms. This is the fundamental definition from which all other possible modes of definition must be derived.

(ii) The square array of symbols that we have called the multiplication table of the group and whose properties were discussed in Chapter 4. Such an array defines a group by specifying all products of the group elements.

(iii) A network of directed segments satisfying the basic properties we have established for the graph of a group. Such a network defines a group by specifying within its structure how any product of group elements corresponds to successive paths on the graph network.

In this chapter, we shall concentrate on showing that there is still another way of defining a group—by means of generators and their relations. We have already had some experience with generators.

Cyclic group C3. We shall begin by examining the simple situation presented by C3, the cyclic group of order 3. This is the group of rotations of an equilateral triangle in its own plane (p. 16). The group C3, as a cyclic group, can be generated by one of its elements, say r, and the three elements can be represented as r, r2, r3 = I.