Integrals of groups

TL;DR

This paper explores the concept of integrals of groups, characterizing which groups have integrals, their sizes, and the conditions under which they exist, providing new classifications and open problems in group theory.

Contribution

It offers a comprehensive analysis of integrals of groups, including finite and infinite cases, and characterizes groups based on their integrability properties and sizes.

Findings

Finite groups with integrals also have finite integrals.

Characterization of natural numbers where all groups are integrable: cubefree and without certain prime divisors.

Abelian groups have integrals of size at most n^{1+o(1)}, but not necessarily proportional to n.

Abstract

An $integral$ of a group $G$ is a group $H$ whose derived group (commutator subgroup) is isomorphic to $G$. This paper discusses integrals of groups, and in particular questions about which groups have integrals and how big or small those integrals can be. Our main results are: (1) If a finite group has an integral, then it has a finite integral. (2) A precise characterization of the set of natural numbers $n$ for which every group of order $n$ is integrable: these are the cubefree numbers $n$ which do not have prime divisors $p$ and $q$ with $q\mid p-1$. (3) An abelian group of order $n$ has an integral of order at most $n^{1+o(1)}$, but may fail to have an integral of order bounded by $cn$ for constant $c$. (4) A finite group can be integrated $n$ times (in the class of finite groups) if and only if it is the central product of an abelian group and a perfect group. There are many other results on such topics as centreless groups, groups with composition length $2$, and infinite groups. We also include a number of open problems.