Some Ways of Organising a Group

Abstract

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 ... a^{n − 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.