This chapter introduces the notion of an abelian group. This is an abstraction that models many different algebraic structures, and yet despite the level of generality, a number of very useful results can be easily obtained.

Definitions, basic properties, and examples

Definition 8.1.An abelian group is a set G together with a binary operation ⋆ on G such that

1. (i) for all a, b, c ∈ G, a ⋆ (b ⋆ c) = (a ⋆ b) ⋆ c (i.e., ⋆ is associative),

2. (ii) there exists e ∈ G (called the identity element) such that for all a ∈ G, a ⋆ e = a = e ⋆ a,

3. (iii) for all a ∈ G there exists a′ ∈ G (called the inverse of a) such that a ⋆ a′ = e = a′ ⋆ a,

4. (iv) for all a, b ∈ G, a ⋆ b = b ⋆ a (i.e., ⋆ is commutative).

While there is a more general notion of a group, which may be defined simply by dropping property (iv) in Definition 8.1, we shall not need this notion in this text. The restriction to abelian groups helps to simplify the discussion significantly. Because we will only be dealing with abelian groups, we may occasionally simply say “group” instead of “abelian group.”