Solvable Groups and the Jordan-Hölder Theorem

We have previously defined the notion of a solvable group in Section 8.1 (Definition 8.1.6). This was done in terms of a sequence of subgroups of the group $G$, viz., the commutator subgroups. In this chapter, we shall give an alternate characterization of solvable groups again in terms of sequences of subgroups. We shall be concerned, in particular, with two types of sequences of subgroups: a normal series and a composition series, and the notion of when two such sequences are equivalent, which will lead to the Jordan-Hölder Theorem. These notions will all be made precise in this chapter. We recall as was observed in the introduction to Chapter 8 that the concept of a solvable group is intimately related to the solvability of a polynomial equation by radicals.