Actions

Actions

This introductory chapter contains the basic notions: actions of groups, orbits, stabilizers, fixed points, and so on. Various examples are given. The Cauchy-Frobenius Lemma is derived, which yields the number of orbits in the case when both the group and the set on which it acts are finite.

In order to prepare the applications of this lemma and its refinements which follow in the next chapters, a detailed description of the conjugacy classes of symmetric and of monomial groups is added.

The paradigmatic actions which we discuss and apply in full detail here and later on are several natural actions on the set Y^X, consisting of all the mappings from X into Y. These actions on Y^X are induced in a natural way by actions of groups on X or on Y. The corresponding orbits are called symmetry classes of mappings and there are many structures in mathematics and sciences which can be defined as symmetry classes of this kind.

The enumeration of symmetry classes of mappings is described in full detail, in order to prepare refinements which are given in the following chapters.

A very simple case of a group action leads to another important enumerative concept, the so-called involution principle. Finally we discuss the enumeration of symmetry classes which consist of injective or surjective mappings only.

• Actions of groups
• Bilateral classes, symmetry classes of mappings
• Finite symmetric groups
• Complete monomial groups
• Enumeration of symmetry classes
• The involution principle
• Special symmetry classes

Actions