Ever-Evolving Groups: An Introduction to Modern Finite Group Theory

This book presents an original approach to the theory of finite groups, placing finite sporadic groups on an equal footing. It provides a nearly comprehensive overview of developments in the study of sporadic groups since the classification of finite simple groups was completed.

Authored by one of the key contributors to these developments, a major theme of the book is the growing role that geometry has played in this story in the form of diagram geometries, amalgams, graph theory and “pushing up”. The chapters interweave various ideas and techniques applicable to all sporadic groups. Many of the results presented—several due to the author and collaborators—appear in book form for the first time. While much of the book describes developments from recent decades, it also includes significant new material, notably on the enigmatic Thompson group and the Monster. The final chapter explores connections to Majorana algebras and discusses some remarkable conjectures.

A valuable addition to the literature on finite simple groups, this book will appeal to a wide audience, from advanced graduate students to researchers in group theory, combinatorics, finite geometry, coding theory, graph theory, and other mathematical fields that use group theory to study symmetries and structures.

Alexander Anatolievich Ivanov is a distinguished Russian mathematician. Born in 1958 in the city of Yaroslavl, he graduated with distinction from the Moscow Institute of Physics and Technology in 1981, where he also received his PhD in 1984. Since that year, he has worked as a senior researcher at the Federal Research Center "Computer Science and Control" of the Russian Academy of Sciences. In 1990, Ivanov delivered a 45-minute invited lecture at the International Congress of Mathematicians in Kyoto, Japan. In 2000, he received a Doctor of Science degree from Moscow State University.

Over the years, Professor Ivanov has held visiting and regular professorial positions at several leading universities worldwide, including the University of Chicago, the University of Michigan, the University of Cambridge, the University of Tokyo, the University of Melbourne, and Imperial College London. At the latter institution, he taught algebra courses for a quarter of a century, which formed the foundation of the present volume.

Professor Ivanov has established numerous groundbreaking results in algebra, combinatorics, and geometry, including Ivanov’s diameter bound for distance-regular graphs, the Y-presentation for the Monster group, and the launch of Majorana theory. He has published over one hundred research papers, five monographs, three collections of papers, and organized a dozen memorable international conferences.

Since September 2024, he is a Professor at the School of Mathematical Sciences, Hebei Normal University, People’s Republic of China.