Integrals of groups II

TL;DR

This paper advances the theory of group integrals by exploring conditions, existence, and properties of integrals across various classes of groups, including finite, abelian, nilpotent, and profinite groups.

Contribution

It provides new criteria for integrability, constructs examples, and investigates the structure of integrals within different group varieties and classes.

Findings

Established bounds on the order of finite integrals.

Proved existence of nilpotent integrals for abelian groups.

Constructed examples of profinite groups without profinite integrals.

Abstract

An $integral$ of a group $G$ is a group $H$ whose commutator subgroup is isomorphic to $G$. This paper continues the investigation on integrals of groups started in the work arXiv:1803.10179. We study: (1) A sufficient condition for a bound on the order of an integral for a finite integrable group and a necessary condition for a group to be integrable. (2) The existence of integrals that are $p$-groups for abelian $p$-groups, and of nilpotent integrals for all abelian groups. (3) Integrals of (finite or infinite) abelian groups, including nilpotent integrals, groups with finite index in some integral, periodic groups, torsion-free groups and finitely generated groups. (4) The variety of integrals of groups from a given variety, varieties of integrable groups and classes of groups whose integrals (when they exist) still belong to such a class. (5) Integrals of profinite groups and a characterization for integrability for finitely generated profinite centreless groups. (6) Integrals of Cartesian products, which are then used to construct examples of integrable profinite groups without a profinite integral. We end the paper with a number of open problems.