Linear Algebraic Groups and Finite Groups of Lie Type

Cambridge Studies in Advanced Mathematics #133

by Gunter Malle, Donna Testerman

Originating from a summer school taught by the authors, this concise treatment includes many of the main results in the area. An introductory chapter describes the fundamental results on linear algebraic groups, culminating in the classification of semisimple groups. The second chapter introduces more specialized topics in the subgroup structure of semisimple groups, and describes the classification of the maximal subgroups of the simple algebraic groups. The authors then systematically develop the subgroup structure of finite groups of Lie type as a consequence of the structural results on algebraic groups. This approach will help students to understand the relationship between these two classes of groups. The book covers many topics that are central to the subject, but missing from existing textbooks. The authors provide numerous instructive exercises and examples for those who are learning the subject as well as more advanced topics for research students working in related areas.

324 pages, Hardcover

First published August 31, 2011

Reviews

[Chris · Rating 5 out of 5]

This book was a great lockdown buddy. It was worth a close read. Humphreys's book warns that the theory can be difficult to "see through" since its most rigorous arguments are long and weave together many different mathematical threads. This book tells a part of this beautiful story of interest to group theorists. Its lucid and tight storytelling makes clear to the novitiate where the algebraic geometry, combinatorics, Lie theory and etc. enter in. In particular, the authors do a great job of emphasizing the parts of the theory (and proofs) which use or illuminate group theory. I was interested in the story of finite simple groups of Lie type and it was satisfying to see, among many other things, how the topological arguments over an algebraically-closed field trickle down to properties of finite subgroups which are the fixed points of some endomorphism. The exercises are well-chosen and enlightening; ideal for (self-)learning.