scholarly journals Finiteness theorems for algebraic groups over function fields

2011 ◽  
Vol 148 (2) ◽  
pp. 555-639 ◽  
Author(s):  
Brian Conrad
Keyword(s):  
Function Fields ◽  
Simple Groups ◽  
Structure Theory ◽  
Reductive Groups ◽  
Compactness Criterion ◽  
Group Schemes ◽  
Global Function Fields ◽  
Finiteness Theorems ◽  
Tamagawa Numbers

AbstractWe prove the finiteness of class numbers and Tate–Shafarevich sets for all affine group schemes of finite type over global function fields, as well as the finiteness of Tamagawa numbers and Godement’s compactness criterion (and a local analogue) for all such groups that are smooth and connected. This builds on the known cases of solvable and semi-simple groups via systematic use of the recently developed structure theory and classification of pseudo-reductive groups.

Introduction

2015 ◽  
Author(s):  
Brian Conrad ◽  
Gopal Prasad
Keyword(s):  
Quadratic Forms ◽  
General Structure ◽  
Root Systems ◽  
Simple Groups ◽  
Reductive Groups ◽  
Central Extensions ◽  
New Techniques ◽  
Characteristic 2 ◽  
Minimal Type

This book deals with the classification of pseudo-reductive groups. Using new techniques and constructions, it addresses a number of questions; for example, whether there are versions of the Isomorphism and Isogeny Theorems for pseudosplit pseudo-reductive groups and of the Existence Theorem for pseudosplit pseudo-simple groups; whether the automorphism functor of a pseudo-semisimple group is representable; or whether there is a Tits-style classification in the pseudo-semisimple case recovering the version due to Tits in the semisimple case. This introduction discusses the special challenges of characteristic 2 as well as root systems, exotic groups and degenerate quadratic forms, and tame central extensions. It also reviews generalized standard groups, minimal type and general structure theorem, and Galois-twisted forms and Tits classification.

scholarly journals Finite simple exceptional groups of Lie type in which all subgroups of odd index are pronormal

2020 ◽  
Vol 23 (6) ◽  
pp. 999-1016
Author(s):  
Anatoly S. Kondrat’ev ◽  
Natalia V. Maslova ◽  
Danila O. Revin
Keyword(s):  
Simple Groups ◽  
Finite Simple Groups ◽  
Exceptional Groups ◽  
Odd Index ◽  
Groups Of Lie Type

AbstractA subgroup H of a group G is said to be pronormal in G if H and {H^{g}} are conjugate in {\langle H,H^{g}\rangle} for every {g\in G}. In this paper, we determine the finite simple groups of type {E_{6}(q)} and {{}^{2}E_{6}(q)} in which all the subgroups of odd index are pronormal. Thus, we complete a classification of finite simple exceptional groups of Lie type in which all the subgroups of odd index are pronormal.

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