1 - Groups and vector spaces

Summary

The material in this chapter should be familiar to the reader, but it is worth reading through it to become familiar with the notation and terminology that is used. We shall not give details; these are given in standard textbooks, such as Mac Lane and Birkhoff [MaB], Jacobson [Jac] or Cohn [Coh].

Groups

A group is a non-empty set G together with a law of composition, a mapping (g, h) → gh from G × G to G, which satisfies:

1. (gh)j = g(hj) for all g, h, j in G (associativity),

2. there exists e in G such that eg = ge = g for all g ∈ G, and

3. for each g ∈ G there exists g^-1 ∈ G such that gg^-1 = g^-1g = e.

It then follows that e, the identity element, is unique, and that for each g ∈ G the inverse g^-1 is unique.

A group G is abelian, or commutative, if gh = hg for all g, h ∈ G. If G is abelian, then the law of composition is often written as addition: (g, h) → g + h. In such a case, the identity is denoted by 0, and the inverse of g by -g.

A non-empty subset H of a group G is a subgroup of G if h₁h₂ ∈ H whenever h₁, h₂ ∈ H, and h^-1 ∈ H whenever h ∈ H.