Use this URL to cite or link to this record in EThOS:
Title: Characterisations of some classes of finite soluble groups
Author: Harman, David Alec
ISNI:       0000 0001 3532 0280
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 1981
Availability of Full Text:
Access from EThOS:
Access from Institution:
The main aim of this thesis is to characterise the so-called "ranked" saturated formations - that is, those saturated formations which are completely determined by the ranks of chief factors of the groups contained in them. This is motivated by recent work of GaschQtz, of Hawkes, and of Heineken. To achieve this characterisation, an area of Clifford theory is developed in Chapter I. In Section 3 of the first chapter, we introduce a method of applying a theorem of Clifford (see I.1.4-) when arbitrary fields of non-zero characteristic are involved without resorting to the theory of representations over algebraic number fields. This method is developed by means of the technique (described in Section 2) of extending the base field of a module to a Galois extension of that field'. In Section k, some corollaries of Theorem 1.3*6 are employed to prove a partial generalisation (Theorem 1.^.2) of a well-known result due, independently, to Swan and Dade. Chapter II is a brief resume of the elementary theory of finite soluble groups. In particular, in Section k the notion of a ranked saturated formation is formally defined, and the relationship between ranked saturated formations and the so-called GaschQtz classes is noted. Chapter III contains the characterisation theorems. Theorem III.2.3 gives arithmetical criteria for a saturated formation to be ranked, and provides a local definition of such formations. Theorem III.2.J proves that the ranked saturated formations are precisely the GaschQtz classes which are formations, and that these are precisely the subgroup- closed GaschQtz classes. The thesis finishes in Chapter IV with a full characterisation of the saturated formations which are completely determined by the absolute ranks of chief factors of the groups contained in them.
Supervisor: Not available Sponsor: Science Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics