Linear Algebraic Groups and Finite Groups of Lie Type

Abstract

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.

A Murnaghan–Nakayama rule for values of unipotent characters in classical groups

“…With respect to an appropriate choice of basis of V we can assume (see [13,Exmp. 6.7]) that the subgroup T of diagonal matrices forms an Fstable maximal torus of G, its elements have the form diag(t 1 , .…”

“…The stabilisers of totally singular spaces are contained in maximal parabolic subgroups [13,Prop. 12.13], but these only have tori of types as excluded by (1).…”

Brauer characters and coprime action

“…Without loss of generality we can assume that some power of F 0 coincides with F . According to Theorem 24.15 of [11] we see that…”

Real homogeneous spaces, Galois cohomology, and Reeder puzzles

“…We consider the group W 0 : (2), and it acts on the set of labelings L(D τ ) via (20). We wish to describe this action explicitly.…”

“…Let H denote the derived subgroup of H 1 . Then by[20, Proposition 12.6] H is a semisimple group with root system R H . Since G is simply connected, by[20, Proposition 12.14] H is simply connected as well.…”