Compact Subgroups of Linear Algebraic Groups

Finite Subgroups of Algebraic Groups

0. Introduction 1105 1. Constructible families 1111 2. Genericity for finite subgroups 1116 3. Finite groups of Lie type 1119 4. Basic nonconcentration estimate 1122 5. Finite subgroups of abelian varieties 1126 6. Orders of conjugacy classes and centralizers 1127 7. Regular semisimple and unipotent elements 1129 8. Minimal unipotent elements 1134 9. Frobenius map 1140 10. Traces in the basic c...

Compact groups as algebraic groups

Aclassic result of [Tannaka:1939], in the formulation of [Chevalley:1946], asserts that every compact subgroup of GLn(C) is the group of real points on an algebraic group defined over R. Chevalley introduced a more algebraic approach to the topic, but his underlying argument is not so different from Tannaka’s. The literature since then has followed one of two threads. One is algebraic (as in Ch...

Characterizing Subgroups of Compact Abelian Groups

We prove that every countable subgroup of a compact metrizable abelian group has a characterizing set. As an application, we answer several questions on maximally almost periodic (MAP) groups and give a characterization of the class of (necessarily MAP) abelian topological groups whose Bohr topology has countable pseudocharacter.

Difference algebraic subgroups of commutative algebraic groups over finite fields

We study the question of which torsion subgroups of commutative algebraic groups over finite fields are contained in modular difference algebraic groups for some choice of a field automorphism. We show that if G is a simple commutative algebraic group over a finite field of characteristic p, ` is a prime different from p, and for some difference closed field (K, σ) the `-primary torsion of G(K)...

Linear Algebraic Groups