Envelopes and covers for groups

TL;DR

This paper bridges module approximation theory and group theory to study envelopes and covers of groups, providing characterizations for specific classes and posing open questions.

Contribution

It introduces a framework connecting module approximation concepts with group theory, enabling the analysis of envelopes and covers for arbitrary groups.

Findings

Characterization results for certain classes of groups

Extension of envelopes and covers to arbitrary groups

Open questions for further research

Abstract

We connect work done by Enochs, Rada and Hill in module approximation theory with work undertaken by several group theorists and algebraic topologists in the context of homotopical localization and cellularization of spaces. This allows one to consider envelopes and covers of arbitrary groups. We show some characterizing results for certain classes of groups, and present some open questions.