Cohomology and the subgroup structure of a finite soluble group
The main topic of this thesis is the discovery and study of a cohomological property of the subgroups called F-normalizers in finite soluble groups; namely, the property that with certain coefficient modules the restriction map in cohomology from a soluble group to its F-normalizers vanishes in non-zero degrees. Chapter 3 is devoted to a proof of this fact It turns out that in some classes of soluble groups the F-normalizers are characterized by this property, and the study of these classes occupies Chapters 4 and 5. Various connections with cohomology and group theory are found; the approach seems to offer some unification of disparate results from the theory of soluble groups. The relation between F-normalizers and cohomology was discovered through study of the work of Jacques Thevenaz on the action of a soluble group on its lattice of subgroups. Chapter 1 is a summary of this work and its background, and is included to provide motivation. A link with the rest of the thesis arises through a new result, in which certain subgroups crucial to Thevenaz's analysis of soluble groups are shown to coincide with their system normalizers. A proof of this is given in Chapter 2, which also contains some miscellaneous results on soluble groups from the class considered by Thevenaz, comprising those groups whose lattices of subgroups are complemented. The problem of characterizing F-normalizers in soluble groups by the results of Chapter 3 is proposed in Chapter 4, and in Chapters 4 and 5 two essentially different approaches to this problem are taken, which lead to partial solutions in different sets of circumstances. In Chapter 4, the first cohomology groups of soluble groups are considered, and an application is given to a proof of a recent theorem of Volkmar Welker described in Chapter 1 on the homotopy type of the partially ordered set of conjugacy classes of subgroups of a soluble group. Another application is to the study of local conjugacy of subgroups of soluble groups, and these are combined in a result which shows that the set of conjugacy classes considered by Welker is homotopy equivalent to an analogous set obtained from local conjugacy classes. In Chapter 5 some known results on the local conjugacy of F-normalizers are exhibited, as evidence for a cohomological characterization of these subgroups. The results are used to study groups of p-length one by a 'local' analysis, whereby the problem of characterizing F-normalizers is translated into a question concerning the action of automorphisms on the cohomology rings of p-groups. In the study of this question a natural place to start is the case of abelian groups, whose cohomology rings are known; calculations in this case lead to results on the F-normalizers of A-groups. The question is then considered for other p-groups, revealing an elegant relationship between the cohomology of p-groups, the theory of varieties, and some well-known results on automorphisms of p-groups.