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Abstract
Let ℱ be a set of finite groups. A finite group G is called an ℱcover if every group in is isomorphic to a subgroup of G. An ℱcover is called minimal if no proper subgroup of G is an ℱcover, and minimum if its order is smallest among all ℱcovers. We prove several results about minimal and minimum covers: for example, every minimal cover of a set of pgroups (for p prime) is a pgroup (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.
Original language  English 

Pages (fromto)  345372 
Journal  Journal of Algebra 
Volume  660 
Early online date  1 Aug 2024 
DOIs  
Publication status  Epub ahead of print  1 Aug 2024 
Keywords
 Cauchy's theorem
 Cayley's theorem
 Simple groups
 Abelian groups
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Dive into the research topics of 'Minimal cover groups'. Together they form a unique fingerprint.Projects
 1 Active

Group Generations From Finite to Infinit: Group Generation: From Finite to Infinite
1/04/23 → 30/09/25
Project: Fellowship