On Integers that are Covering Numbers of Groups

The covering number of a group G, denoted by is the size of a minimal collection of proper subgroups of G whose union is G. We investigate which integers are covering numbers of groups. We determine which integers 129 or smaller are covering numbers, and we determine precisely or bound the covering...

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Bibliographic Details
Published inExperimental mathematics Vol. 31; no. 2; pp. 425 - 443
Main Authors Garonzi, Martino, Kappe, Luise-Charlotte, Swartz, Eric
Format Journal Article
LanguageEnglish
Published Taylor & Francis 27.07.2022
Subjects
covering number
Primary 20D60
primitive group
Secondary 20B15
subgroup cover
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Summary:The covering number of a group G, denoted by is the size of a minimal collection of proper subgroups of G whose union is G. We investigate which integers are covering numbers of groups. We determine which integers 129 or smaller are covering numbers, and we determine precisely or bound the covering number of every primitive monolithic group with a degree of primitivity at most 129 by introducing effective new computational techniques. Furthermore, we prove that, if is the family of finite groups G such that all proper quotients of G are solvable, then is infinite, which provides further evidence that infinitely many integers are not covering numbers. Finally, we prove that every integer of the form where and q is a prime power, is a covering number, generalizing a result of Cohn.
ISSN:1058-6458
1944-950X
DOI:10.1080/10586458.2019.1636425