Carr, Ben (1998) Model subgroups of finite soluble groups. PhD thesis, University of Warwick.

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Abstract

In this thesis we begin the study of finite groups possessing a model subgroup, where a model subgroup H of a finite group G is defined to be a subgroup satisfying
〖1H〗^(↑G)=∑_(x∊∕π(G))▒X
We show that a finite nilpotent group possesses a model subgroup if and only if it is abelian and that a Frobenius group with Frobenius complement C and Frobenius kernel N possesses a model subgroup if and only if
(a) N is elementary abelian of order r".
(b) C is cyclic of order (r" — 1 )/(rd — 1), for some d dividing n.
(c) The finite field F=Frn has an additive abelian subgroup HF of order rd satisfying NormF/K(HF) =K, where K=Frd.
We then go on to conjecture that a finite soluble group G possessing a model subgroup is either metabelian or has a normal subgroup N such that G/N is a Frobenius group with cyclic Frobenius complement of order 2" +1 and elementary abelian Frobenius kernel of order 22". We consider a series of cases that need to be excluded in order to prove the conjecture and present some examples that shed light on the problems still to be overcome.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Finite groups, Solvable groups, Mathematics
Official Date:	December 1998
Dates:	Date Event December 1998 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Hawkes, Trevor O.,1936-
Sponsors:	Science and Engineering Research Council (Great Britain)
Extent:	v, 176 leaves
Language:	eng

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