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Subgroup Coverings of Some Sporadic Groups

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Subgroup Coverings of Some Sporadic Groups View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Combinatorial Theory, Series A 113 (2006) 1204–1213 www.elsevier.com/locate/jcta Note Subgroup coverings of some sporadic groups P.E. Holmes School of Mathematics and Statistics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK Received 2 October 2004 Available online 1 December 2005 Abstract A set of proper subgroups is a covering for a group if their union is the whole group. Determining the size of a smallest covering is an open problem for many simple groups. For some of the sporadic groups, we find subgroup coverings of minimal cardinality. For others we specify the isomorphism types of subgroups in a smallest covering and use graphs to calculate bounds for its size. © 2005 Elsevier Inc. All rights reserved. Keywords: Group theory; Sporadic groups 1. Introduction Let G be a non-cyclic finite group and let S be a set of proper subgroups of G. Then S is a subgroup covering or simply a covering for G if G = S. Cohn [4] defined σ(G) to be the minimal number of proper subgroups needed to cover the group G. We follow Tomkinson [12] in calling a covering of G consisting of σ(G) subgroups a minimal covering. Some results on σ(G) for different types of groups G have been proved. For example, Bryce and Serena [2] gives a characterisation of groups having a minimal covering consisting of abelian subgroups and Bubbolini [3] considers groups that can be covered with only two conjugacy classes of subgroups, or even one conjugacy class and only one other subgroup. Praeger [9] looks at the coverings {H α: α ∈ A} where Inn(G) A Aut(G). In a paper on covering soluble groups, Tomkinson [12] showed that σ(G)= pa + 1 where pa is the size of the smallest chief factor group that has more than one complement. In this paper he showed interest in knowing σ(G) for simple groups G. Bryce, Fedri and Serena begun this project in [1] where they calculated σ(G)for some linear groups. Their main result was that E-mail address: [email protected]. 0097-3165/$ – see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jcta.2005.09.006 Note / Journal of Combinatorial Theory, Series A 113 (2006) 1204–1213 1205 = 1 + = 1 + + =∼ σ(G) 2 q(q 1) when q is even and σ(G) 2 q(q 1) 1 when q is odd where G PSL2(q), PGL2(q),SL2(q) or GL2(q) and q = 2, 5, 7, 9. Maroti [8] looks at the symmetric and alternating n−2 n−1 groups and shows that σ(An) 2 for n = 7 or 9 and σ(Sn) 2 . Lucido [7] investigates the Suzuki groups and shows that σ(Sz q) = q2(q2 + 1)/2. This paper also gives a survey of the subject. No investigation of the finite simple groups is complete without looking at the twenty six sporadic simple groups. This paper considers seven of them. 1.1. Notation ATLAS notation is used throughout the paper. Here we give a brief description of it. Many non-abelian finite simple groups are denoted by a single capital letter, together with some numbers if necessary. For example, PSLn(q) is written as Ln(q) and the Mathieu groups n n are named M11,...,M24. Elementary abelian groups of order p are denoted by p (or just p if n is 1). A split extension, or semi-direct product, of H by K is written H : K, while a non-split extension is written H ˙K. Conjugacy classes are named by the orders of their elements and a letter. They are written in descending order of centraliser size. For example, Janko’s second group J2 has two classes of involutions. Centralisers of two different types of involution have orders 1920 and 240, so we call the classes 2A and 2B. 1.2. Results Table 1 gives minimal coverings that are unions of full conjugacy classes of subgroups. First note that to find σ(G) it is clear that we only need consider the number of maximal subgroups of G needed to cover all maximal cyclic subgroups of G.LetC be a conjugacy class of cyclic subgroups. When an element of C appears in more than one conjugate of a maximal subgroup H then a minimal covering might not be a union of conjugacy classes of subgroups. In this case we define a certain graph, ΓG(H, C). Then we use the maximal size of an independent set in ΓG(H, C) to calculate bounds for σ(G). (A set of vertices I is independent if no two vertices in I are joined by an edge.) The definition of these graphs is in Section 3 and more details are given in Section 4. Table 2 gives bounds for coverings that are not unions of conjugacy classes of maximal sub- groups. In these cases we will show that any minimal covering may only contain groups of the isomorphism types listed in the second column of Table 2. We will also show that the total num- ber of subgroups in a minimal covering lies between the bounds given in the third column. Note that σ(M11) was calculated independently by Maroti in [8]. Table 1 G Maximal subgroups used σ(G) M11 L2(11),M10 23 4 M22 M21,L2(11),2 :A6 771 4 M23 L3(4) :2or2 :A7,A8, 23 : 11 41079 C Ly 2˙A11,3˙M L : 2, 37 : 18, 67 : 22, G2(5) 112845655268156 O’N 42˙L3(4) :21,J1,L2(31) 36450855 1206 Note / Journal of Combinatorial Theory, Series A 113 (2006) 1204–1213 Table 2 G Maximal subgroups used Bounds on σ(G) 3 J1 19 : 6, 2 :7:3,D6 × D10,L2(11) 5165, 5415 C M LM22,2˙A8,U4(3) 24541, 24553 Section 2 gives the results of Table 1, Section 3 the crucial result that ties subgroup coverings to independent subsets of graphs, Section 4 the methods used to obtain the results in Table 2, and Section 5 the proofs of the assertions of Table 2. 2. Some coverings This section gives the results in Table 1. All the coverings found in this section are unions of conjugacy classes of maximal subgroups. 2.1. Finding small coverings Recall that to find σ(G) it is clear that we only need consider the number of maximal subgroups of G needed to get all maximal cyclic subgroups of G. The power maps between conjugacy classes of generators of the maximal cyclic subgroups are stored in the ATLAS [5] and in the computer algebra package GAP [10]. Lists of all conjugacy classes of maximal subgroups of most sporadic groups can be found in the ATLAS [5] and in GAP [10]. If H is a subgroup of G, then each conjugacy class of H consists of elements from one conjugacy class of G. The map between the classes of H and the classes of G is called the fusion map. The fusion maps between the conjugacy classes of the simple group and those of its maximal subgroups are found in GAP. This information is enough to determine which sets of conjugacy classes of subgroups of G are coverings of G. We can also use the class fusion maps to count the number of conjugates of a subgroup H that contain a given element c of a conjugacy class C. This is given by (G : NG(H ))|H ∩ C|/|C|. It is usually easy to determine which subgroups are best to use to cover a given conjugacy class. But when the best subgroups for one conjugacy class are a bad choice for a different conjugacy class then the situation is more complicated. We wrote a G AP program to identify some groups where we do not have this sort of conflict. This program outputs a minimal covering where one can easily be found. When exact sizes cannot be produced but it is possible to state which conjugacy classes of subgroups are involved in a minimal covering, then this partial information is given to the user. In the next subsection we give an outline of our GAP program and in Section 2.3 we work through an example by hand to show what the program does. 2.2. Our program The input for the program is the GAP format character table for a finite non-abelian simple group G. This is either read directly from the GAP character table library or, if not available, input directly by the user. As well as the character table itself, these records also include a list of the group’s conjugacy classes of maximal subgroups and the fusion maps between the conjugacy classes of each maximal subgroup and those of G. Note / Journal of Combinatorial Theory, Series A 113 (2006) 1204–1213 1207 First the program uses the list of power maps between conjugacy classes to determine a list of conjugacy classes C. This list contains the name of exactly one G-class of generators for each maximal cyclic group. The program also initialises an empty covering, S. Next the program looks through the list C and uses the fusion maps to note which of the classes of maximal subgroups contain elements from each class in C. If no element of a conju- gacy class X appears in any conjugate of a maximal subgroup M, then we say M is impossible for X.
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