ABSTRACT
In this chapter we introduce two families of groups: nilpotent and solvable. One can think of these groups as generalizations of abelian groups. The main reason we cover this theory is for the application of solvable groups to answer the question if there exists a formula for finding the roots of a polynomial of degree five or more, otherwise known as solvability by radicals. Solvable groups are named as such because of this connection. In Section 6.1 we remind the reader of some important subgroups as well as introduce some additional important subgroups. In Section 6.2 we discuss certain chains of subgroups of a group which we use in Section 6.3 to define nilpotent and solvable groups.
