Minimal cover groups

Research output: Contribution to journalArticleResearchpeer review

Authors

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

External Research Organisations

  • University of St. Andrews
  • University of Birmingham
  • University of Jiroft
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Details

Original languageEnglish
Pages (from-to)345-372
Number of pages28
JournalJournal of algebra
Volume660
Early online date23 Jul 2024
Publication statusE-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 {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

Cite this

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 journalArticleResearchpeer review

Cameron, PJ, Craven, D, Dorbidi, HR, Harper, S & Sambale, B 2024, 'Minimal cover groups', Journal of algebra, vol. 660, pp. 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. Advance online publication. 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 Dec 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 ; Vol. 660. pp. 345-372.
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