I’m pleased to announce I’ve republished my MSci thesis on Finite Supersoluble Groups, together with my PhD-era lecture notes on Finitary Permutation Groups, combined into one book.

Two Essays on Group Theory is free to download on Leanpub and as a hardback book from Amazon.

Here is the preface from the book.

Preface

This book contains two essays I wrote when I was studying mathematics at the end of the 20th century. The first essay is my MSci thesis on supersoluble groups. The second essay is a set of lecture notes on finitary permutation groups.

The first essay is a review of results about finite supersoluble groups. In chapter 2 we introduce the idea of a supersoluble group and we investigate its connection with other similar concepts such as solubility and nilpotency. In chapter 3 we look at supersoluble series and present some forms of these which are common to all supersoluble groups.

The main result of chapter 4 is a conjugacy theorem of Philip Hall on Hall π-subgroups in finite groups. In chapter 5 we present some characterisation theorems for finite supersoluble groups, including theorems of Huppert, Kramer and Iwasawa. We also give a necessary and sufficient condition for finite supersolubility in terms of the converse of Lagrange’s Theorem.

Chapter 6 presents some miscellaneous results on supersoluble groups.

I submitted the first essay in March 1997 as my MSci thesis. In 1998, I converted the thesis from Word to LaTeX. In 2026, I converted it to Leanpub Markua. I’ve made a few corrections.

The second essay is a set of notes produced whilst giving a workshop to fellow PhD students and colleagues at the Combinatorics Study Group at Queen Mary, University of London. The subject is group theory, but because the groups are of permutations, I could get away with passing it off as combinatorics. I aimed the notes at an audience who had seen some basic permutation group theory, but little abstract group theory.

A finitary permutation group is a natural generalisation of a finite permutation group. The structure of a transitive finitary permutation group is not complicated when its degree is infinite. Here we study primitivity, following P. M. Neumann’s work in the 1970s. We also study generalised solubility conditions on these groups.

The study group was a lively forum and group theory was often a topic. Actually, the forum used to continue being lively in the Old Globe afterwards and on one occasion, we drank the pub dry. Anyway, I digress.

I cannot remember when I produced this essay but it’s likely to have been during the second year of my PhD. In fact I had trouble finding the LaTeX files. I converted it to Leanpub Markua in June 2026. The astute reader will notice that in two places I refer to notes by Peter Cameron. I’m afraid any hope of finding a reference to these notes has been lost in the mists of time. If I recall correctly, he’d given a lecture on permutation groups around the same time as my session.

I used Claude to check both essays; it found several textual inconsistencies and conversion errors.

I would like to take this opportunity to thank my PhD supervisor Bert Wehrfritz for his help and advice during my MSci and PhD. Peter Kropholler was available for discussions over beer during the course of writing the first essay. Peter Cameron, Leonard Soicher and Rosemary Bailey helped at the Old Globe, and provided the study group environment for the second essay. My undergraduate tutor Sue McKay was instrumental in my mathematical development, as were Don Collins, Stephen Donkin and Ian Chiswell. David Arrowsmith was there as a reminder that mathematics could be applied as well. Ruth and Elaine in the office were also supportive. I haven’t written all the names down here but I look back at the eight years at Queen Mary with great fondness.

Chris Pinnock
June 2026