Algebra I: A Basic Course in Abstract Algebra

Chapter 5

GroupDefinition and Examples

In this chapter we shall study different algebraic structures with one binary operation and the relationship amongst them. The simplest algebraic structure is a groupoid. We begin with a few definitions.

5.1 Definition of Group

Definition 5.1.   A non-empty set G equipped with a binary operation * is called a groupoid, that is, a * b ∈ G ∀ a, b ∈ G. This is also referred to as: G is closed with respect to *.

Definition 5.2.   A non-e mpty set G equipped with a binary operation * is called a semigroup if * is associative, i.e.

 

a * (b * c)=(a * b) * c ∀ a, b, ∈ G.          (5.1)

 

Definition 5.3.   A non-empty set G equipped with a binary operation * is called a monoid if

1. * is associative, i.e.

    

   a * (b ...