Chapter 2 - The Structure of Groups

Abstract

This chapter discusses the structure of groups. The associative law of equation implies that in any product of three or more elements, no ambiguity arises if the brackets are removed completely. They can be inserted freely around any chosen subset or subsets of elements in the product, provided the order of elements is unchanged. A concise criterion for a subset of a group to be a subgroup is provided. The Rearrangement Theorem implies that in the multiplication table of a finite group, every element of the group appears once in every row and once in every column. It is shown that the same coset is formed starting from any member of the coset. All members of a coset appear on an equal footing, so that any member of the coset can be taken as the coset representative that labels the coset and from which the coset can be constructed. The homomorphic and isomorphic mappings are elaborated. The direct products and semidirect products of groups are also analyzed.