Model theoretic connected components of groups

TL;DR

This paper explores the model theoretic connected components of groups, demonstrating that NIP groups have a smallest invariant subgroup with bounded index, extending Shelah's theorem to various algebraic structures.

Contribution

It generalizes Shelah's theorem by establishing the existence of a smallest invariant subgroup with bounded index in NIP groups, including rings and fields.

Findings

NIP groups have a smallest invariant subgroup with bounded index

Extension of Shelah's theorem to additive and multiplicative groups of rings

Application to infinite fields and other algebraic structures

Abstract

We give a general exposition of model theoretic connected components of groups. We show that if a group G has NIP, then there exists the smallest invariant (over some small set) subgroup of G with bounded index (Theorem 5.3). This result extends theorem of Shelah. We consider also in this context the multiplicative and the additive groups of some rings (including infite fields).