On Integers that are Covering Numbers of Groups

Abstract

The covering number of a group G, denoted by σ(G), 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 F 1 is the family of finite groups G such that all proper quotients of G are solvable, then N − {σ(G) : G ∈ F 1 } is infinite, which provides further evidence that infinitely many integers are not covering numbers. Finally, we prove that every integer of the form (q m − 1)/(q − 1), where m = 3 and q is a prime power, is a covering number, generalizing a result of Cohn.2010 Mathematics Subject Classification. Primary 20D60; Secondary 20B15.

The covering numbers of rings

“…Refer to Theorem 1.3(4) for a summary of the results on A(n, q 1 , q 2 ). Our strategy is based on the approach used to prove [15,Theorem 1.7], where it is shown that for n = 2, the group AGL(n, q) has covering number (q n+1 − 1)/(q − 1). However, our formulas for σ(A(n, q 1 , q 2 ))-and the methods used to establish themare more complicated than in the group case.…”

“…Proof. The proof is essentially identical to that of [15,Lemma 5.1] and [31,Lemma 3.1] with the word "group" changed to the word "ring," but it is included for the sake of completeness. Let D and Π be as in the statement of the lemma, and assume that c(M) 1 for all maximal subrings not in D; the proof with strict inequalities is analogous.…”

“…As noted above, no group has covering number 7, and other natural numbers exist that are not the covering number of a group (the next smallest examples being 11 [10] and 19 [14]). All the integers N 129 that are not the covering number of a group were determined in [15], which also contains a good summary of the history and recent development of the related work for groups. It is not known whether there are infinitely many positive integers that are not the covering number of a group.…”

On the Intersection Numbers of Finite Groups

“…Theorem 3.4. (Garonzi [10]; Garonzi, Kappe and Swartz [11]) The following list identifies exactly which integers less than 130 that are not covering numbers: The corollary below proves that for each k ∈ Z + , there exists a group G such that ι(G) = k.…”

“…The remaining case was confirmed by Detomi and Lucchini [6], who proved that there is no group G with σ(G) = 11. Additionally, the combined results of Garonzi, Kappe, and Swartz [10,11] classified every integer less than 130 that is a covering number of some nontrivial group. More generally, the structure of groups G containing no normal nontrivial subgroup N such that σ(G/N) = σ(G) was investigated by Detomi and Lucchini [6].…”