Chapter 6
Monoids and groups


  1. Binary operations

  2. Monoids

  3. Invertibility in monoids

  4. Groups

  5. Cyclic groups

  6. Cosets

  7. Exercises

  8. Summary of Chapter 6

In the previous five chapters we have considered several sets with operations, like addition and multiplication, defined on them. Such an enriched set is often called a structure. In this chapter we start with a more systematic approach to structures. The title of this chapter refers to the two most important ones we shall deal with here.

Section 6.1
Binary operations
  1. Arity
  2. Associativity
  3. Semi-groups
  4. Monoids
  5. Commutativity
Section 6.2
Monoids
  1. Direct product
  2. Submonoid
  3. Free monoid
  4. Morphisms
  5. Cyclic monoids
Section 6.3
Invertibility in Monoids
  1. Inverse
  2. Properities of inverse
  3. Euler indicator
Section 6.4
Groups
  1. Subgroups
  2. Subgroups generated by subsets
  3. Intersections
  4. Some special subgroups
  5. Morphisms
  6. Kernel and image
Section 6.5
Cyclic groups
  1. Order of an element
  2. Characterizing cyclic groups
Section 6.6
Cosets
  1. The notion
  2. Lagrange
  3. Normal subgroups