Actions of Groups, Topological Groups and Semigroups

Abstract

Chapter 3 discusses actions of semigroups, groups, topological groups, and Lie groups. Each element of a group determines a permutation on a set under a group action. For a topological group action on a topological space, this permutation is a homeomorphism and for a Lie group action on a differentiable manifold it is a diffeomorphism. Group actions are used in the proofs of the Counting Principle, Cayley’s Theorem, Cauchy’s Theorem and Sylow Theorems for finite groups. The counting principle is used to determine the structure of a finite group. These groups arise in the Sylow Theorems and in the description of finite abelian groups. The orbit spaces obtained by topological group actions, discussed in this chapter, are very important in topology and geometry. For example, n-dimensional real and complex projective spaces are obtained as orbit spaces. Finally, semigroup actions are applied to theoretical computer science yielding state machines which unify computer science with mainstream mathematics.