On the covering number of small symmetric groups and some sporadic simple groups

Abstract

ABSTRACT. A set of proper subgroups is a covering for a group if its union is the whole group. The minimal number of subgroups needed to cover G is called the covering number of G, denoted by σ(G). Determining σ(G) is an open problem for many non-solvable groups. For symmetric groups S n , Maróti determined σ(S n ) for odd n with the exception of n = 9 and gave estimates for n even. In this paper we determine σ(S n ) for n = 8, 9, 10 and 12. In addition we find the covering number for the Mathieu group M 12 and improve an estimate given by Holmes for the Janko group J 1 .

On Integers that are Covering Numbers of Groups

“…A GAP check shows that the only possibility is X ∼ = Aut(PSL(2, 27)), in which case X/N ∼ = C 6 and ℓ X (N ) = 28. In this case, Lemma 4.1 implies that either σ(X) σ * (X) + 2 < 2σ * (X), or, for one of the g i 's in this lemma, N, g i ∼ = PGL (2,27), and so σ * (X) σ(PGL(2, 27)) − 1 = 378 holds, a contradiction to σ * (X) 129. Thus σ(X) < 2σ * (X).…”

“…Theorem 1.1. The integers between 26 and 129 which are not covering numbers are 27,34,35,37,39,41,43,45,47,49,51,52,53,55,56,58,59,61,66,69,70,75,76,77,78,79,81,83,87,88,89,91,93,94,95,96,97,99,100,101,103,105,106,107,109,111,112,113,115,116,117,118,119,120,123,124,125. Theorem 1.1 follows from Theorem 4.5, Proposition 6.1, and Table 6.…”

“…Computationally, there are two main methods that we use. The first is a method developed in [27], where a program in GAP creates a system of equations that can be solved (either partially or totally) by the linear optimization software GUROBI [21]. This method, which we refer to as Algorithm KNS, is detailed in Section 5.1.…”

On the covering number of symmetric groups having degree divisible by six

“…We now need to show that C is in fact a minimal cover. We define Π to be the set of all elements of S 18 with cycle structure one of (18), (7,11), (1,7,10), (3,7,8), (4,7,7), or (5,6,7). Note that these elements are partitioned among the subgroups in C. We index the classes of maximal subgroups of C as follows: we let the subgroups isomorphic to S 9 wr S 2 be M −1 , the subgroup isomorphic to A 18 be M 0 , and the subgroups isomorphic to S i × S 18−i be M i for i = 1, 3, 4, 5.…”

The covering number of $M_{24}$

“…The covering number of M 12 was determined by L. C. Kappe, D. Nikolova-Popova, and E. Swartz in [8].…”