 
Summary: Chapter 4
Groups
The theory of groups is a branch of math
ematics in which one does something to
something and then compares the result
with the results of doing the same thing
to something else, or something else to the
same thing.
 James Newman
This chapter introduces groups, an abstract cat
egory of objects that are foundational to the rest of
algebra and, for that matter, to the rest of mathemat
ics. Applications of groups appear in fields as diverse
as art, crystallography, and quantum mechanics. We
begin with the basic definitions and some examples
of groups.
4.1 Definitions and Examples
Definition 4.1 A group is a set G together with a
binary operation that obey the following three ax
ioms.
