# Groups with Locally Modular Homogeneous Pregeometries are Commutative

## TL;DR

This paper proves that groups with locally modular homogeneous pregeometries are necessarily commutative, extending the known result for strongly minimal groups to a broader class.

## Contribution

It establishes that groups with locally modular homogeneous pregeometries are commutative, generalizing the classical result for strongly minimal groups.

## Key findings

- Groups with locally modular homogeneous pregeometries are commutative.
- Extends the known commutativity result from strongly minimal groups to a wider class.
- Supports the conjecture that certain geometric conditions imply group commutativity.

## Abstract

It is well known that strongly minimal groups are commutative. Whether this is true for various generalisations of strong minimality has been asked in several different settings (see Hyttinen [2002], Maesono [2007], Pillay and Tanovi\'c [2011]). In this note we show that the answer is positive for groups with locally modular homogeneous pregeometries.