Subgroups of Finite Index in Profinite Groups

One way to view Theorem 1.1 is as a statement that the algebraic structure of a finitely generated profinite group somehow also encodes the topological structure. That is, if one wishes to know the open subgroups of a profinite group G, a topological property, one must only consider the subgroups of G of finite index, an algebraic property. As profinite groups are compact topological spaces, an open subgroup of G necessarily has finite index. Thus it is also possible to begin with a subgroup of G having a particular topological property (the subgroup is open) and deduce that this subgroup must also have a particular algebraic property (the subgroup has finite index). The proof of Theorem 1.1 is quite extensive, and requires the classification of finite simple groups. However, if one restricts attention to a smaller class of groups, the result can be done in a fairly straightforward manner. Suppose that G is a finite group having a normal series 1 = Gl ⊆ Gl−1 ⊆ . . . ⊆ G1 ⊆ G0 = G such that Gi/Gi+1 is nilpotent for all 0 ≤ i < l. Then we say that G belongs to the class N . Note that finite nilpotent groups all belong to N1, while finite supersolvable groups all belong to N2 since the commutator subgroup of a supersolvable group is nilpotent. The theorem we aim to prove in this document is the following.