Elementary Group Theory

Abstract

There are two important reasons for considering groups in this book. On the one hand, the notion of a group exhibits a fundamental mathematical structure that is found, for example, in rings, fields, vector spaces, and modules, in which one interprets the inherent addition as a law of composition. All groups of this type are commutative or, as we also will say, abelian, referring to the mathematician N. H. Abel. On the other hand, there are groups originating from another source, such as the so-called Galois groups related to the work of E. Galois. These groups will be of central interest for us, serving as a key tool for the investigation of algebraic equations. From a simplified point of view, Galois groups are permutation groups, i.e., groups whose elements describe bijective transformations (self-maps) on sets like {1, …, n}.