Summary of Chapter 5

Overview of sections

1. Symmetric groups
2. Cycles
3. Alternating groups
4. Exercises

Overview of contents

The main subject is the bijections of the set {1, 2, ..., n} to itself, the so-called permutations of 1, 2, ..., n. Permutations may be multiplied (product of permutations) and we may take their inverses (the inverse of a permutation). In both cases the result is again a permutation. The set of all permutations of 1, 2, ..., n is denoted by S_n. We deal with the following subjects.

• Presentations of a permutation:
  • as an ordered row;
  • as a 2×n matrix;
  • by a permutation matrix;
  • as a product of disjoint cycles (NB, it is a theorem that this is possible);
  • as a product of transpositions (NB, it is a theorem that this is possible);
  • by a graph.
  For this the following terms are important:
  • (m-)cycle, 2-cycle or transposition;
  • the fixed points of a permutation;
  • disjoint cycles.
• Calculating with permutations in the different presentations.
• The sign of a permutation with the rules: even × even = even,    odd × even = odd,    odd × odd = even.
• The subset of even permutations A_n in S_n.

Permutations are an important tool for describing symmetries (for example, those of a cube); this aspect will also be dealt with in Chapter 8.