Chapter 8
Permutation groups


  1. Permutation groups

  2. Orbits

  3. Order

  4. Automorphisms

  5. Quotient groups

  6. Small groups

  7. Exercises

  8. Summary of Chapter 8

Groups have been introduced in Chapter 6 as abstract sets with some operations. However, groups often appear as transformations mapping a set onto itself. For example, in the group of real invertible n×n matrices, each element determines a bijective linear map Rn -> Rn. Such `group actions' on a set enable us to analyze the group in a concrete setting. But it is also a means of unveiling the symmetries of structures on that set.

In this chapter, we look into the way a group G can be represented by letting it act on a set or structure.

Section 8.1
Permutation groups
  1. The notion
  2. Each group is a permutation group
  3. Constructions
Section 8.2
Orbits
  1. The notion
  2. Stabilizers
  3. Transitive actions are actions on cosets
Section 8.3
Order
  1. Basis
  2. Schreier trees
  3. Stabilizer determination
Section 8.4
Automorphisms
  1. The notion
  2. Finite fields
Section 8.5
Quotient groups
  1. The notion
  2. First isomorphism theorem
Section 8.6
Small groups
  1. Preliminaries
  2. Order at most 11