Semi-Riemannian Geometry

Appendix B Abstract Algebra

B.1. Groups

A group is a pair (G, ∘), where G is a set and ∘ : G × G → G is a map, called multiplication, such that for all f, g, h in G:

1. [Gl] Multiplication is associative:
   
2. [G2] There is an element id _G in G, called the identity, such that
   
3. [G3] There is an element g ⁻¹ in G, called the inverse of g, such that
   

Usually g ∘ h is abbreviated to gh. It is easily shown that the identity element and the inverse of an element are unique. It is standard practice to refer to G as a group, leaving the multiplication map understood. We say that G is commutative if multiplication is commutative; that is, gh = hg for all g, h in G. In that situation, it is usual to employ additive notation, with the group denoted by (G, +), the identity by 0 _G or 0, and the inverse of g by −g .

A subset G ^′ of G is said to be a subgroup of G if: (i) G ^′ is a group in its own right (when multiplication on G is restricted to G ^′ ), and (ii) G ^′ has the same identity element as G. We say that a group is finite if it is finite as a set.

Theorem B.1.1

If {G _α  : α ∈ A} is a collection of subgroups of G, then ∩ _α ∈ A G _α is a subgroup ...