Integrals of groups. II
Date
24/04/2024Metadata
Show full item recordAbstract
An integral of a group G is a group H whose derived group (commutator subgroup) is isomorphic to G. This paper continues the investigation on integrals of groups started in the work [1]. We study: -- A sufficient condition for a bound on the order of an integral for a finite integrable group (Theorem 2.1) and a necessary condition for a group to be integrable (Theorem 3.2). -- The existence of integrals that are p-groups for abelian p-groups, and of nilpotent integrals for all abelian groups (Theorem 4.1). -- 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 (Section 5). -- 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 (Sections 6 and 7). -- Integrals of profinite groups and a characterization for integrability for finitely generated profinite centreless groups (Section 8.1). -- Integrals of Cartesian products, which are then used to construct examples of integrable profinite groups without a profinite integral (Section 8.2). We end the paper with a number of open problems.
Citation
Araujo , J , Cameron , P J , Casolo , C , Matucci , F & Quadrelli , C 2024 , ' Integrals of groups. II ' , Israel Journal of Mathematics . https://doi.org/10.48550/arXiv.2008.13675 , https://doi.org/10.1007/s11856-024-2610-4
Publication
Israel Journal of Mathematics
Status
Peer reviewed
ISSN
0021-2172Type
Journal article
Description
Funding: The first author was funded by national funds through the FCT - Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UIDB/00297/2020, UIDP/00297/2020 (Center for Mathematics and Applications) and PTDC/MAT/PUR/31174/2017. The first, second and fourth authors gratefully acknowledge the support of the Fundação para a Ciência e a Tecnologia (CEMAT-Ciências FCT projects UIDB/04621/2020 and UIDP/04621/2020); and the fourth author gratefully acknowledge the support of the Universit‘a degli Studi di Milano–Bicocca (FA project ATE-2017-0035 “Strutture Algebriche”).Collections
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