This thesis is in two parts. The first is a study of soluble profinite groups with the maximal condition (max) and mirrors the development of the fundamental properties of polycyclic groups. Two of the main results are a profinite version of Hall's criterion for nilpotence and the existence of nilpotent almost-supplements for the Fitting subgroup. It is shown that the automorphism group of a soluble profinite group with max itself has max and that a soluble profinite group has max if and only if each of its subnormal subgroups of defect at most 2 has max. Quantitative versions of these results are also obtained. The second part derives a new characterization of abstract and profinite branch groups. This is achieved by examining the subnormal subgroup structure of just non-(virtually abelian) groups and a partial classification of this larger class is obtained. Weak branch groups are shown to satisfy no abstract group laws and to contain many abstract free subgroups in the profinite case. The notion of a branch group is weakened to that of a generalized branch group. Associated with each generalized branch group is its structure graph, and the circumstances under which this graph is a tree are characterized.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:269887 |
| Date | January 2002 |
| Creators | Hardy, Philip David |
| Publisher | University of Birmingham |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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