The covering number of $M_{24}$

Abstract

A finite cover C of a group G is a finite collection of proper subgroups of G such that G is equal to the union of all of the members of C. Such a cover is called minimal if it has the smallest cardinality among all finite covers of G. The covering number of G, denoted by σ(G), is the number of subgroups in a minimal cover of G. In this paper the covering number of the Mathieu group M24 is shown to be 3336.

On the Intersection Numbers of Finite Groups

“…Proof. The order-42 maximal subgroups M 1 = (1, 2, 3, 4, 5, 6, 7), (2,6,5,7,3,4) and M 2 = (1, 2, 7, 3, 5, 6, 4), (2,6,5,4,7,3) of (3,4), (5,6) of S 7 , which isomorphic to the elementary abelian 2-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]. There have also been investigations into the values of σ(G) for certain nonsolvable groups G (see [3,7,9,14,15,18,19,20,23]), but in general establishing the covering number of a nonsolvable group remains a topic of ongoing research. The focus of this article is an investigation into a dual problem to that of covering numbers of groups, which involves intersections of maximal subgroups of groups.…”

“…(1), σ 8 (1), σ 12 (1), σ 16 (1), σ 20 (1) σ 24 (1), σ 28 (1), σ 32 (1), σ 36 (1), σ 40 (1) = (1, 5, 9, 2, 6, 10) (3,7,11,4,8), 7),…”

“…. , 11} are defined by α 0 = 1, σ(1), σ 2 (1), σ 3 (1), σ 4 (1), σ 5 (1) σ 6 (1), σ 7 (1), σ 8 (1), σ 9 (1), σ 10 (1) = (1, 2, 3, 4, 5, 6) (7,8,9,10,11) and β 0 = 1, σ(1) σ 6 (1), σ 7 (1) = (1, 2) (7,8).…”

On Integers that are Covering Numbers of Groups

“…Element distribution in O −(8,2) It is clear from Tables 31 and 32 that no maximal subgroup contains elements from more than one of the classes cl 15,1 , cl 15,2 , or cl15,3 . A minimal cover for each class consists of at least ⌈135/2⌉ subgroups, and so at least 204 subgroups are needed in any cover.…”