Local Sylow theory of totally disconnected, locally compact groups

TL;DR

This paper introduces the concept of local Sylow subgroups in totally disconnected, locally compact groups, and explores their role in defining p-localisation and analyzing group properties.

Contribution

It defines local Sylow subgroups and p-localisation for these groups, providing new tools to study their structure and properties.

Findings

Defined local Sylow subgroups as maximal pro-p subgroups of open compact subgroups.

Established the relationship between scale and modular functions of G and its p-localisation.

Extended the framework to locally virtually prosoluble groups using local Sylow bases.

Abstract

We define a local Sylow subgroup of a totally disconnected, locally compact group G to be a maximal pro-p subgroup of an open compact subgroup of G. We use these subgroups to define the p-localisation of G, a locally virtually pro-p group which maps continuously and injectively to G with dense image, and describe the relationship between the scale and modular functions of G and those of its p-localisation. In the case of locally virtually prosoluble groups, we consider all primes simultaneously using local Sylow bases.