Harman, David Alec (1981) Characterisations of some classes of finite soluble groups. PhD thesis, University of Warwick.

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Abstract

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.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Finite groups, Galois theory
Official Date:	1981
Dates:	Date Event 1981 UNSPECIFIED
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Hawkes, Trevor O., 1936-
Sponsors:	Science Research Council (Great Britain)
Extent:	viii, 165, [3] leaves
Language:	eng

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