Some Ways of Organising a Group

The number a3 is often defined as the product of three factors each equal to a. Another way of defining it is to say that a3 is the result of multiplying the identity element 1 by a three times in succession. This has the advantage of making sense of the rather baffling expression a0. To multiply 1 by a zero times naturally leaves it unchanged suggesting that a0 = 1. It is convenient to adopt this notation in the study of groups and to describe any cyclic group of order n as consisting of the elements 1, a, a2, a3 ... an − 1. For example the non-zero residue classes (mod 7), may be written 30, 31, 32, 33, 34, 35 or otherwise 1, 3, 2, 6, 4, 5. The multiplication table of the elements written in this order shows the characteristic diagonal pattern of a cyclic group.