Partial complements in finite soluble groups

Abstract

Let G be a finite group with normal subgroup N. Let K be a subgroup of G. We say that K is a partial complement of N in G if N and K intersect trivially. There are two main results in this work. The first main result arises from analysing when each partial complement of N in G is contained in a complement of N in G when G is a finite soluble group, N is the product of minimal normal subgroups, N is complemented and all the complements of N in G are conjugate. We show that each partial complement of N in G is contained in a complement of N in G if and only if N is projective. The next natural question is: if N is non-projective, which partial complements are contained in a complement of N in G? We say that a cyclic p-partial complement is a partial complement that is cyclic and its order is a power of p. We establish exactly which cyclic p-partial complements are contained in a conjugate of H.

So the next question is: if the partial complement is a non-cyclic p-partial complement, how do we know that is contained in a conjugate of H? This is a difficult question and because of restriction on representation theory, we have needed to restrict H to be in the class of groups that are nilpotent p'-groups by p-groups. To answer this question, we use the first cohomology group. The first cohomology group is the number of conjugacy classes of complements to N in G. That is, if we have a partial complement K such that the first cohomology group vanishes then we know there is only one conjugacy class of complements of N in NK and therefore K is in a conjugate of H. The second main result finds exactly when the first cohomology group vanishes. This is a sufficient condition for a partial complement to be contained in a complement of N in G. -- provided by Candidate.