Minimal cover groups

authored by: Peter J. Cameron, David Craven, Hamid Reza Dorbidi, Scott Harper, Benjamin Sambale
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 {Z_q,Z_r} 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.
Organisation(s): Institute of Algebra, Number Theory and Discrete Mathematics
External Organisation(s): University of St. Andrews
University of Birmingham
University of Jiroft
Type: Article
Journal: Journal of algebra
Volume: 660
Pages: 345-372
No. of pages: 28
ISSN: 0021-8693
Publication date: 15.12.2024
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Algebra and Number Theory
Electronic version(s): https://doi.org/10.1016/j.jalgebra.2024.06.038 (Access: Open)

BibTeX

@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 = dec,
day = "15",
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.",
}