 # Finite Group Theory: Odd Primes and CC-Subgroups

McLean, Joseph Francis (1985) Finite Group Theory: Odd Primes and CC-Subgroups. MSc(R) thesis, University of Glasgow.

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## Abstract

The majority of this thesis comprises a survey of existing knowledge. Let G be a finite group and suppose that M is a subgroup of G such that (i)1<M<G, and (ii)for all x∈M#,CG(x)≤M. Such a subgroup M is called a CC-subgroup the concept to which this thesis is predominantly devoted. Following a brief introduction. Chapter 11 consists of a survey of the known results on the odd prime structure of finite groups. This survey is split into three sections as follows. The first gives an account of the development of a unified theory for characterising groups with CC-subgroups of order divisible by three. Section II introduces the twin ideas of closure and homogeneity, concluding with a theorem which has Important applications later in Chapter V. Section ill consists of a straightforward listing of the remaining odd prime structure results. Chapter III is the theoretical base of the thesis, contributing all the major results which are required before proceeding. Section II of this chapter is itself an integral part of the survey, being a systematic exposition of basic CC-subgroup theory. Chapter IV is a discussion on the various techniques and proofs Involved In Chapter II, giving a readable yet rigorous explanation of the theory. Chapter V highlights more recent, and more general, results involving CC-subgroups, giving detailed proofs, and sets the scene for the final chapter. Chapter VI consists of two sections. Section I is given over entirely to the statement and proof of a single theorem which completely classifies groups containing CC-subgroups, a simple corollary of which Initiates Section II, an outline of the search for CC-subgroups of the finite simple groups. This section, and the thesis, ends with four tables that give as complete a list as possible of the Information currently available on the CC-subgroups of the simple groups.

Item Type: Thesis (MSc(R)) Masters Adviser: Neil K Dickson Theoretical mathematics 1985 Enlighten Team glathesis:1985-76428 Copyright of this thesis is held by the author. 19 Nov 2019 14:31 19 Nov 2019 14:31 http://theses.gla.ac.uk/id/eprint/76428 View Item