A Study in Locally Compact Groups

TL;DR

This paper introduces and explores the structure of locally compact near abelian groups, connecting them to prime number theory and providing classifications of certain group types using advanced topological and algebraic tools.

Contribution

It develops a comprehensive theory of locally compact near abelian groups, extending existing group theory concepts and providing classifications of quasihamiltonian and modular groups.

Findings

Classification of locally compact topologically quasihamiltonian groups

Retrieval and extension of Mukhin's classification of modular groups

Analysis of the structure of locally compact near abelian groups

Abstract

The class of locally compact near abelian groups is introduced and investigated as a class of metabelian groups formalizing and applying the concept of scalar multiplication. The structure of locally compact near abelian groups and its close connections to prime number theory are discussed and elucidated by graph theoretical tools. These investigations require a thorough reviewing and extension to present circumstances of various aspects of the general theory of locally compact groups such as -the Chabauty space of closed subgroups with its natural compact Hausdorff topology, -a very general Sylow subgroup theory for periodic groups including their Hall systems, -the scalar automorphisms of locally compact abelian groups, -the theory of products of closed subgroups and their relation to semidirect products, and -inductively monothetic groups are introduced and classified. As applications, firstly, a complete classification is given of locally compact topologically quasihamiltonian groups, which has been initiated by F. K\"ummich, and, secondly, Yu. Mukhin's classification of locally compact topologically modular groups is retrieved and further illuminated.