Affine algebraic groups with periodic components

TL;DR

This paper characterizes periodic components in affine algebraic groups, linking them to automorphisms with finite fixed points, and explores their extensions and applications to normalizers in simple algebraic groups.

Contribution

It provides a new characterization of periodic components in affine algebraic groups and analyzes their extensions and applications to group theory.

Findings

Periodic components are characterized by automorphisms with finite fixed points.

Connected groups can have finite extensions with periodic components.

Applications to normalizers of maximal tori in simple algebraic groups.

Abstract

A connected component of an affine algebraic group is called periodic if all its elements have finite order. We give a characterization of periodic components in terms of automorphisms with finite number of fixed points. It is also discussed which connected groups have finite extensions with periodic components. The results are applied to the study of the normalizer of a maximal torus in a simple algebraic group.