## Details

Original language | English |
---|---|

Pages (from-to) | 345-372 |

Number of pages | 28 |

Journal | Journal of algebra |

Volume | 660 |

Early online date | 23 Jul 2024 |

Publication status | E-pub ahead of print - 23 Jul 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 {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.

## Keywords

- Abelian groups, Cauchy's theorem, Cayley's theorem, Simple groups

## ASJC Scopus subject areas

- Mathematics(all)
**Algebra and Number Theory**

## Cite this

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**Minimal cover groups.**/ Cameron, Peter J.; Craven, David; Dorbidi, Hamid Reza et al.

In: Journal of algebra, Vol. 660, 15.12.2024, p. 345-372.

Research output: Contribution to journal › Article › Research › peer review

*Journal of algebra*, vol. 660, pp. 345-372. https://doi.org/10.1016/j.jalgebra.2024.06.038

*Journal of algebra*,

*660*, 345-372. Advance online publication. https://doi.org/10.1016/j.jalgebra.2024.06.038

}

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 -