Cohomology and the subgroup structure of a finite soluble group
Wilde, Thomas Stephen (1992) Cohomology and the subgroup structure of a finite soluble group. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b1411265~S15
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.
|Item Type:||Thesis or Dissertation (PhD)|
|Subjects:||Q Science > QA Mathematics|
|Library of Congress Subject Headings (LCSH):||Solvable groups, Homology theory|
|Official Date:||May 1992|
|Institution:||University of Warwick|
|Theses Department:||Mathematics Institute|
|Supervisor(s)/Advisor:||Hawkes, Trevor O., 1936- ; Mond, D. (David)|
|Sponsors:||Science and Engineering Research Council (Great Britain) (SERC)|
|Extent:||iv, 131 p.|
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