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 | Published - 15 Dec 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.
Keywords
- Abelian groups, Cauchy's theorem, Cayley's theorem, Simple groups
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
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In: Journal of algebra, Vol. 660, 15.12.2024, p. 345-372.
Research output: Contribution to journal › Article › Research › peer review
}
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 - 2024/12/15
Y1 - 2024/12/15
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 -