We start this chapter by abstracting the properties of SA to which we drew attention in qn 1.34 and use these properties to define a group in a way which does not depend on permutations or geometry and in which the elements of the group need not even be functions. For two functions, composition has been presumed to be the method of combination. For two numbers, addition or multiplication are possible methods of combination. In general the phrase ‘binary operation’ is used to describe the combination of two elements to make one element. Some immediate consequences of the group axioms are identified in qns 5–9.

Concurrent reading: Green, chapters 4 and 5, Fraleigh, sections 2, 3, 6 and 7.

Axioms

So far every group we have considered has been a subgroup of a symmetric group. We now widen the definition of a group to include those sets, G, with a binary operation (·), not just composition of functions, which satisfy the four conditions which we insisted on for symmetric groups, namely

1. closure If a, b ∈ G, then a·b ∈ G.

2. associativity If a, b, c ∈ G, then (a·b)·c = a·(b·c).

3. identity There exists e ∈ G such that a·e = e·a = a for all a ∈ G.

4. inverses For each a ∈ G, there exists an a-1 such that a·a-1 = a-1·a = e.

5. 1 […]