ABSTRACT
We begin this chapter with a brief discussion of an axiomatic structure that may be new to you. After set theory, it is one of the most pervasive in all of mathematics.
Definition 11.1.1 Let G be a set, and let P W G G ! G be a function. We call the ordered pair .G;P / a group if the following axioms are satisfied (following custom, we shall usually write P.g; h/ as g h): 1. Associativity If g; h; k 2 G, then g .h k/ D .g h/ kI 2. Identity Element There is a distinguished element e 2 G such that,
for all g 2 G; e g D g e D g. 3. Multiplicative Inverse For each g 2 G there is an element h 2 G
such that g h D h g D e. It is common to denote the inverse element specified in Axiom 3 by g1, and we shall do so in what follows.
