# Centralizers in pseudo-finite groups

## TL;DR

This paper investigates the structure of pseudo-finite groups, showing they are finite-by-abelian-by-finite under certain automorphism conditions and proving they always contain an infinite abelian subgroup, with implications in model theory.

## Contribution

It provides a new model-theoretic proof of a periodic group result and characterizes pseudo-finite groups with specific automorphisms.

## Key findings

- Pseudo-finite groups with a definable involutory automorphism fixing finitely many elements are finite-by-abelian-by-finite.
- Every pseudo-finite group contains an infinite abelian subgroup.
- A model-theoretic proof of a classical periodic group result is established.

## Abstract

The role of finite centralizers of involutions in pseudo-finite groups is analyzed. It is shown that a pseudo-finite group admitting a definable involutory automorphism fixing only finitely many elements is finite-by-abelian-by-finite. As a consequence, we give a model-theoretic proof of a result for periodic groups due to Hartley and Meixner. Furthermore, it is shown that any pseudo-finite group has an infinite abelian subgroup.