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

23.07.2024

Publication status

E-pub ahead of print

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)