Large localizations of finite groups

TL;DR

This paper constructs specific group localizations transforming the Mathieu group into larger groups with near-abelian properties, exploring how localizations affect group structures.

Contribution

It introduces new examples of localizations in group theory that significantly alter the size and properties of the Mathieu group.

Findings

Constructed localizations map M11 to arbitrarily large groups

These groups are 'abelian up to finitely many generators'

The work advances understanding of property preservation under localizations

Abstract

We construct examples of localizations in the category of groups which take the Mathieu group $M_{11}$ to groups of arbitrarily large cardinality which are ``abelian up to finitely many generators''. The paper is part of a broader study on the group theoretic properties which are or are not preserved by localizations.