On regular groups and fields

TL;DR

This paper explores properties of regular groups and fields, focusing on their algebraic closure, generic types, and conjugacy classes, and constructs examples illustrating these concepts in model theory.

Contribution

It introduces new constructions of groups with unique conjugacy class properties and analyzes the algebraic closure of regular fields beyond the generically stable case.

Findings

A regular, non-generically stable field's generic type has unbounded orbit in finite extensions.

Existence of a group of size ω₁ with one non-trivial conjugacy class and countable centralizers.

Demonstrates that certain classical groups with one conjugacy class are not quasi-minimal.

Abstract

Regular groups and fields are common generalizations of minimal and quasi-minimal groups and fields, so the conjectures that minimal or quasi-minimal fields are algebraically closed have their common generalization to the conjecture that each regular field is algebraically closed. Standard arguments show that a generically stable regular field is algebraically closed. Let $K$ be a regular field which is not generically stable and let $p$ be its global generic type. We observe that if $K$ has a finite extension $L$ of degree $n$, then $p^{(n)}$ has unbounded orbit under the action of the multiplicative group of $L$. Known to be true in the minimal context, it remains wide open whether regular, or even quasi-minimal, groups are abelian. We show that if it is not the case, then there is a counter-example with a unique non-trivial conjugacy class, and we notice that a classical group with one non-trivial conjugacy class is not quasi-minimal, because the centralizers of all elements are uncountable. Then we construct a group of cardinality $\omega_1$ with only one non-trivial conjugacy class and such that the centralizers of all non-trivial elements are countable.