Local-global invariants of finite and infinite groups: around Burnside from another side

TL;DR

This paper explores the Shafarevich-Tate set of a group, examining its various forms, properties, and conjectural relationships with other local-global invariants in the context of finite and infinite groups.

Contribution

It provides an overview of the different incarnations and properties of the Shafarevich-Tate set, highlighting its significance in the study of local-global invariants of groups.

Findings

Multiple incarnations of the Shafarevich-Tate set are discussed.

Relationships with other local-global invariants are explored, some conjectural.

The paper offers a comprehensive overview of the set's properties and significance.

Abstract

This expository essay is focused on the Shafarevich-Tate set of a group $G$. Since its introduction for a finite group by Burnside, it has been rediscovered and redefined more than once. We discuss its various incarnations and properties as well as relationships (some of them conjectural) with other local-global invariants of groups.