Minimal cover groups

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

Autoren

  • Peter J. Cameron
  • David Craven
  • Hamid Reza Dorbidi
  • Scott Harper
  • Benjamin Sambale

Externe Organisationen

  • University of St. Andrews
  • University of Birmingham
  • University of Jiroft
Forschungs-netzwerk anzeigen

Details

OriginalspracheEnglisch
Seiten (von - bis)345-372
Seitenumfang28
FachzeitschriftJournal of algebra
Jahrgang660
Frühes Online-Datum23 Juli 2024
PublikationsstatusElektronisch veröffentlicht (E-Pub) - 23 Juli 2024

Abstract

Let F be a set of finite groups. A finite group G is called an F-cover if every group in F is isomorphic to a subgroup of G. An F-cover is called minimal if no proper subgroup of G is an F-cover, and minimum if its order is smallest among all F-covers. We prove several results about minimal and minimum F-covers: for example, every minimal cover of a set of p-groups (for p prime) is a p-group (and there may be finitely or infinitely many, for a given set); every minimal cover of a set of perfect groups is perfect; and a minimum cover of a set of two nonabelian simple groups is either their direct product or simple. Our major theorem determines whether {Zq,Zr} has finitely many minimal covers, where q and r are distinct primes. Motivated by this, we say that n is a Cauchy number if there are only finitely many groups which are minimal (under inclusion) with respect to having order divisible by n, and we determine all such numbers. This extends Cauchy's theorem. We also define a dual concept where subgroups are replaced by quotients, and we pose a number of problems.

ASJC Scopus Sachgebiete

Zitieren

Minimal cover groups. / Cameron, Peter J.; Craven, David; Dorbidi, Hamid Reza et al.
in: Journal of algebra, Jahrgang 660, 15.12.2024, S. 345-372.

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

Cameron, PJ, Craven, D, Dorbidi, HR, Harper, S & Sambale, B 2024, 'Minimal cover groups', Journal of algebra, Jg. 660, S. 345-372. https://doi.org/10.1016/j.jalgebra.2024.06.038
Cameron, P. J., Craven, D., Dorbidi, H. R., Harper, S., & Sambale, B. (2024). Minimal cover groups. Journal of algebra, 660, 345-372. Vorabveröffentlichung online. https://doi.org/10.1016/j.jalgebra.2024.06.038
Cameron PJ, Craven D, Dorbidi HR, Harper S, Sambale B. Minimal cover groups. Journal of algebra. 2024 Dez 15;660:345-372. Epub 2024 Jul 23. doi: 10.1016/j.jalgebra.2024.06.038
Cameron, Peter J. ; Craven, David ; Dorbidi, Hamid Reza et al. / Minimal cover groups. in: Journal of algebra. 2024 ; Jahrgang 660. S. 345-372.
Download
@article{5c2d37cece964fd3add6c464aa7ec4fa,
title = "Minimal cover groups",
abstract = "Let F be a set of finite groups. A finite group G is called an F-cover if every group in F is isomorphic to a subgroup of G. An F-cover is called minimal if no proper subgroup of G is an F-cover, and minimum if its order is smallest among all F-covers. We prove several results about minimal and minimum F-covers: for example, every minimal cover of a set of p-groups (for p prime) is a p-group (and there may be finitely or infinitely many, for a given set); every minimal cover of a set of perfect groups is perfect; and a minimum cover of a set of two nonabelian simple groups is either their direct product or simple. Our major theorem determines whether {Zq,Zr} has finitely many minimal covers, where q and r are distinct primes. Motivated by this, we say that n is a Cauchy number if there are only finitely many groups which are minimal (under inclusion) with respect to having order divisible by n, and we determine all such numbers. This extends Cauchy's theorem. We also define a dual concept where subgroups are replaced by quotients, and we pose a number of problems.",
keywords = "Abelian groups, Cauchy's theorem, Cayley's theorem, Simple groups",
author = "Cameron, {Peter J.} and David Craven and Dorbidi, {Hamid Reza} and Scott Harper and Benjamin Sambale",
note = "Publisher Copyright: {\textcopyright} 2024 The Author(s)",
year = "2024",
month = jul,
day = "23",
doi = "10.1016/j.jalgebra.2024.06.038",
language = "English",
volume = "660",
pages = "345--372",
journal = "Journal of algebra",
issn = "0021-8693",
publisher = "Academic Press Inc.",

}

Download

TY - JOUR

T1 - Minimal cover groups

AU - Cameron, Peter J.

AU - Craven, David

AU - Dorbidi, Hamid Reza

AU - Harper, Scott

AU - Sambale, Benjamin

N1 - Publisher Copyright: © 2024 The Author(s)

PY - 2024/7/23

Y1 - 2024/7/23

N2 - Let F be a set of finite groups. A finite group G is called an F-cover if every group in F is isomorphic to a subgroup of G. An F-cover is called minimal if no proper subgroup of G is an F-cover, and minimum if its order is smallest among all F-covers. We prove several results about minimal and minimum F-covers: for example, every minimal cover of a set of p-groups (for p prime) is a p-group (and there may be finitely or infinitely many, for a given set); every minimal cover of a set of perfect groups is perfect; and a minimum cover of a set of two nonabelian simple groups is either their direct product or simple. Our major theorem determines whether {Zq,Zr} has finitely many minimal covers, where q and r are distinct primes. Motivated by this, we say that n is a Cauchy number if there are only finitely many groups which are minimal (under inclusion) with respect to having order divisible by n, and we determine all such numbers. This extends Cauchy's theorem. We also define a dual concept where subgroups are replaced by quotients, and we pose a number of problems.

AB - Let F be a set of finite groups. A finite group G is called an F-cover if every group in F is isomorphic to a subgroup of G. An F-cover is called minimal if no proper subgroup of G is an F-cover, and minimum if its order is smallest among all F-covers. We prove several results about minimal and minimum F-covers: for example, every minimal cover of a set of p-groups (for p prime) is a p-group (and there may be finitely or infinitely many, for a given set); every minimal cover of a set of perfect groups is perfect; and a minimum cover of a set of two nonabelian simple groups is either their direct product or simple. Our major theorem determines whether {Zq,Zr} has finitely many minimal covers, where q and r are distinct primes. Motivated by this, we say that n is a Cauchy number if there are only finitely many groups which are minimal (under inclusion) with respect to having order divisible by n, and we determine all such numbers. This extends Cauchy's theorem. We also define a dual concept where subgroups are replaced by quotients, and we pose a number of problems.

KW - Abelian groups

KW - Cauchy's theorem

KW - Cayley's theorem

KW - Simple groups

UR - http://www.scopus.com/inward/record.url?scp=85199947060&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2024.06.038

DO - 10.1016/j.jalgebra.2024.06.038

M3 - Article

AN - SCOPUS:85199947060

VL - 660

SP - 345

EP - 372

JO - Journal of algebra

JF - Journal of algebra

SN - 0021-8693

ER -