CHAPTER ONE Definitions and Examples of Groups 1A From the abstract, axiomatic point of view that prevails today, one can argue that group theory is, in some sense, more primitive than most other parts of algebra and, indeed, the group axioms constitute a subset of the axiom systems that define the other algebraic objects considered in this book. Things we learn about groups, therefore, will often be relevant to our study ofmodules, rings, andfields.In addition, group theory has considerable indirect connection to these other areas. (The most striking example of this is probably the use of Galois groups to study fields.) It is largely for these reasons that we begin this book on algebra with an extensive study of group theory. (If the whole truth were told, the fact that the author's primary research interest and activity are in group theory would be seen as relevant, too.) The subject we call "algebra" was not born abstract. In its youth, algebra was the study of concrete objects such as polynomials, rather than of things defined by axiom systems. In particular, early group theory was concerned with groups of mappings, known as "transformation groups." (In the early literature, for instance, the elements of a group were referred to as its "operations") For at least two reasons, we begin our study of group theory by (temporarily) adopting this nineteenth-century point of view. First, mappings of one kind or another are ubiquitous throughout algebra (and most of the rest of mathematics, too) and so it makes sense to begin with them. Furthermore, some of the most interesting examples of groups are best constructed and visualized as transformation groups. We begin our study of mappings with some notation and definitions. (It is this author's belief that mathematics at its best consists of theorems and examples. Definitions are often dull, although they are a necessary evil, especially near the beginning of an expository work. We pledge that the balance of theorems and examples versus definitions will become more favorable as the reader progresses through the book.) 3
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