Model subgroups of finite soluble groups

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