ABSTRACT
In the final two chapters of the book — this one and the next — I am going to introduce you to the theory of groups. This theory forms part of a vast area known as “abstract algebra”. In abstract algebra we study certain basic systems in which we have a set, together with rules for combining any two elements of the set to get another element of the set; these rules are subject to various clearly defined assumptions, called “axioms.” In group theory, as we shall see, there is just one rule for combining elements, and there are four axioms. You should not think that these axioms were thought up by some clever person who one sunny day sat down and wrote them down. Rather, they emerged over a long period — many different cases of what have come to be known as groups were studied in the eighteenth and nineteenth centuries, but it was not until late in the nineteenth century that the notion of an abstract group was introduced.
