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Integrals of groups II

João Araújo, Peter J. Cameron, Carlo Casolo, Francesco Matucci, Claudio Quadrelli

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.

Integrals of groups II

Abstract

An of a group is a group whose commutator subgroup is isomorphic to . 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 -groups for abelian -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.

Paper Structure

This paper contains 24 sections, 40 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Bounding the order of an integral
  3. Step 1
  4. Step 2
  5. Step 3
  6. Towards a characterization of integrable groups
  7. $p$-integrals
  8. Integrals of abelian groups
  9. Nilpotent integrals
  10. Finitely integrable abelian groups
  11. Periodic groups
  12. Ulm--Kaplansky invariants
  13. Torsion-free groups
  14. Finitely generated integrals
  15. Varieties of groups
  16. ...and 9 more sections

Key Result

Theorem \oldthetheorem

Suppose there is a function $F$ from finite groups to natural numbers such that, if $G$ is an integrable finite group, then $F(G)$ is a bound for the exponent of the centre of some integral $H$ of $G$. Then there is a function $F^*$ from finite groups to natural numbers suth that, if $G$ is an integ

Theorems & Definitions (83)

  • Theorem \oldthetheorem
  • Proof 1
  • Example \oldthetheorem
  • Theorem \oldthetheorem: Eick eick
  • Theorem \oldthetheorem
  • Proof 2
  • Theorem \oldthetheorem
  • Proof 3
  • Theorem \oldthetheorem: Burnside burnside
  • Example \oldthetheorem
  • ...and 73 more