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:If a finite group has an integral, then it has a finite integral.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 | p − 1.An abelian group of order n has an integral of order at most n1+o(1), but may fail to have an integral of order bounded by cn for constant c.A finite group can be integrated n times (in the class of finite groups) for every n if and only if it is a 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

Araujo, J., Cameron, P., Casolo, C., Matucci, F. (2019). Integrals of groups. ISRAEL JOURNAL OF MATHEMATICS, 234(1), 149-178 [10.1007/s11856-019-1926-y].

Integrals of groups

Matucci F.
Co-primo
2019

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:If a finite group has an integral, then it has a finite integral.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 | p − 1.An abelian group of order n has an integral of order at most n1+o(1), but may fail to have an integral of order bounded by cn for constant c.A finite group can be integrated n times (in the class of finite groups) for every n if and only if it is a 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
Articolo in rivista - Articolo scientifico
Derived subgroup, commutator subgroup
English
2019
234
1
149
178
reserved
Araujo, J., Cameron, P., Casolo, C., Matucci, F. (2019). Integrals of groups. ISRAEL JOURNAL OF MATHEMATICS, 234(1), 149-178 [10.1007/s11856-019-1926-y].
File in questo prodotto:
File Dimensione Formato  
Araujo-2019-Isr J Math-VoR.pdf

Solo gestori archivio

Descrizione: Article (uncorrected proof)
Tipologia di allegato: Publisher’s Version (Version of Record, VoR)
Licenza: Tutti i diritti riservati
Dimensione 394.26 kB
Formato Adobe PDF
394.26 kB Adobe PDF   Visualizza/Apri   Richiedi una copia
Araujo-2019-Isr J Math-VoR.pdf

Solo gestori archivio

Descrizione: Article
Tipologia di allegato: Publisher’s Version (Version of Record, VoR)
Licenza: Tutti i diritti riservati
Dimensione 327.19 kB
Formato Adobe PDF
327.19 kB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/261883
Citazioni
  • Scopus 3
  • ???jsp.display-item.citation.isi??? 3
Social impact