\(\newcommand{\identity}{\mathrm{id}} \newcommand{\notdivide}{\nmid} \newcommand{\notsubset}{\not\subset} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\gf}{\operatorname{GF}} \newcommand{\inn}{\operatorname{Inn}} \newcommand{\aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\cis}{\operatorname{cis}} \newcommand{\chr}{\operatorname{char}} \newcommand{\Null}{\operatorname{Null}} \newcommand{\transpose}{\text{t}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \setcounter{chapter}{-1}\)

Chapter3Finite Groups; Subgroups

As we saw in the previous chapter, one of the most complete (though unsophisticated) ways to understand a group is to study its Cayley table that, for a group of \(n\) elements, contains \(n^2\) entries. Of course, this is only possible to do if \(n\) is finite! So we will begin --- and indeed, we'll spend most of our time --- studying the properties of groups whose sets of elements are finite, called finite groups. In this chapter, we'll set down some basic notions about finite groups, and begin to take a closer look at finite groups in search of smaller groups (subgroups) residing within them that can help us better to understand their structure.