From Subdirect Sums to Virtuality

TL;DR

This paper generalizes classical results on subgroups of direct sums by introducing the concept of virtuality, providing new insights and techniques in module extensions and diagrammatic structures within modular representation theory.

Contribution

It extends Remak's theorem to infinite direct sums and develops the notion of virtuality in module diagrams, offering new methods for analyzing subgroup structures.

Findings

Generalization of Remak's theorem to infinite direct sums

Introduction of virtuality in module extensions and diagrams

New techniques for analyzing subgroup structures and homomorphisms

Abstract

We start by an original investigation on subgroups of (even infinite) direct sums in the first 4 sections, that largely generalizes Remak's known theorem; inspired by that general picture we have elsewhere extended this elementary "virtual" diagrammatic situation (in diagrammatic length 2 meaning set-theoretic fixation of vertices) by generalizing to the notion of "virtuality" in module extensions and diagrams in modular representation theory. Our first approach starts with an appropriately defined equivalence relation, which is precisely what allows for treating the confusing case of multiple factors, thus giving a deeper insight into the structure of such subgroups. Several applications and new techniques arising from that approach are examined, even ones concerning basic properties of homomorphisms, extending well-known elementary ones.