Summary of Chapter 6

Overview of sections

1. Binary operations
2. Monoids
3. Invertibility in monoids
4. Groups
5. Cyclic groups
6. Cosets
7. Exercises

Overview of contents

In this chapter we have started the study of algebraic structures, i.e., sets with operations defined on them. We have introduced two main structures, monoids and groups. A monoid is a triple (M,*,e) consisting of a set M, an associative binary operation * of M and a unit element e with respect to this operation. A group is a quadruple (G,*,e, x->x^-1), where (G,*,e) is a monoid and x^-1 is an inverse for x. For both structures we have discussed

• constructions,
• substructures and
• morphisms.

If G is a group and H is a subgroup, then the left cosets of H in G partition the group G (and similarly for the right cosets). This leads to the important theorem of Lagrange: The order of H divides the order of G.