Rank 2 Primitive Geometries of the Mathieu Group

Applied Mathematical Sciences, Vol. 8, 2014, no. 128, 6349 - 6357 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.48642

All Rank 2 Primitive Geometries

of the Mathieu Group M11

Nayil Kilic

Sinop University, Faculty of Arts and Sciences Department of Mathematics, 57000, Sinop, Turkey

Copyright c 2014 Nayil Kilic. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we determine all rank 2 primitive geometries for the Mathieu group M11 for which object stabilizers are maximal subgroups.

Mathematics Subject Classiﬁcation: 20D08, 51E10, 05C25

Keywords: Mathieu groups, Steiner system, group geometries

1. Introduction One of the major open questions nowadays in finite simple groups is to find a unified geometric interpretation of all of the finite simple groups. The theory of buildings due to Jacques Tits answer partially this question by associating a geometric object to all of the finite simple groups except the Alternating groups and the sporadic groups. Since 1970, Francis Buekenhout introduced diagram geometries, allowing more residues than just generalized polygons and started building geometries for the sporadic groups. In that spirit, he classified with Dehon and Leemans all primitive geometries for the Mathieu group M11 (see [3]). Since 1993, several mathematician, including Oliver Bauduin, Francis Bueken- hout, Philippe Cara, Michel Dehon, Xavier Miller, Koen Vanmeerbeek and Dimitri Leemans have classified geometries under the following assumptions. The geometries obtained must be firm, residually connected, flag-transitive, residually weakly primitive (RWPRI) and they must satisfy the intersection property of rank two. Classification for certain groups were given in [[4], [5], 6350 Nayil Kilic

[7], [8], [11]]. These results rely partially on the use of algorithms described by Dehon [10] for the computer algebra package CAYLEY [7] and translated later in MAGMA [1]. In [3], using a Cayley program, Authors get all firm, residually connected geometries whose rank two residues satisfy the intersection property on which M11 acts flag-transitively, and in which the stabilizer of each element is a maximal subgroup of M11. Under these conditions they get 8 rank two geometries. In this paper, we give the list of all rank two geometries for M11 that are firm, residually connected and flag-transitive. We get 33 geometries of rank 2. We list these geometries explicitly by their diagram. These results were checked by using a series of MAGMA [1] programs. The paper is organized as follows. In section 2, we recall the basic definitions needed in order to understand this paper. In section 3, we give the list of geometries we obtained. Finally, in section 4, we give the diagram of rank 2 geometries for M11.

2. Definitions and Notation We begin by reviewing geometries and some standard notation. A geometry is a triple (Γ,I,?) where Γ is a set, I an index set and ? a symmetric incidence relation on Γ which satisfy

. (i) Γ = ∪ Γi; and i∈I (ii) if x ∈ Γi, y ∈ Γj(i, j ∈ I) and x ? y, then i 6= j.

The elements of Γi are called objects of type i, and |I| is the rank of the geometry Γ (as is usual we use Γ is place of the triple (Γ,I,?)). A flag F of Γ is a subset of Γ in which every two element of F are incident. The rank of F is |F |, the corank of F is |I\F | and the type of F is {i ∈ I|F ∩ Γi 6= ∅}. A chamber of Γ is a flag of type I. All geometries we consider are assumed to contain at least one flag of rank |I|. The automorphism group of Γ, AutΓ, consists of all permutations of Γ which preserve the sets Γi and the incidence relation ?. Let G be a subgroup of AutΓ. We call Γ a flag transitive geometry for G if for any two flags F1 and F2 of Γ having the same type, there exists g g ∈ G such that F1 = F2. For ∆ ⊆ Γ, the residue of ∆, denoted Γ∆, is defined to be {x ∈ Γ|x ? y for all y ∈ ∆}. We call Γ residually connected provided that the incidence graph of each residue of rank ≥ 2 is a connected graph. We call Γ firm provided that every flag of rank |I| − 1 is contained in at least two chambers. The diagram of a firm, residually connected, flag-transitive geometry Γ is a complete graph K, whose vertices are the elements of the set of type I of Γ, provided with some additional structure which is further described as follows. To each vertex i ∈ I, we attach the order si which is |ΓF | − 1 where F is any flag of type I\{i}, and the number ni of varieties of type i, which is the index of Gi in G, and the subgroup Gi. To every edge {i, j} of K, we associate All rank 2 primitive geometries of the Mathieu group M11 6351 three positive integers dij, gij and dji where gij (the gonality) is equal to half the girth of the incidence graph of a residue ΓF of type {i, j}, and dij (resp. dji), the i − diameter (resp. j − diameter) is the greatest distance from some fixed i− element ( resp. j− element ) to any other element in ΓF . On a picture of the diagram, this structure will often be depicted as follows.

d gij d ij ji OO si sj ni nj Gi Gj B = Gij

For each i ∈ I choose an xi ∈ Γi and set Gi = StabG(xi). Let F={Gi : i ∈ I}. We now deﬁne a geometry Γ(G, F) where the objects of type i in Γ(G, F) are the right cosets of Gi in G and for Gix and Gjy (x, y ∈ G, i, j ∈ I) Gix ? Gjy whenever Gix ∩ Gjy 6= ∅. Also by letting G act upon Γ(G, F) by right multiplication we see that Γ(G, F) is a ﬂag transitive geometry for G. Moreover Γ and Γ(G, F) are isomorphic geometries for G. So we shall be ∼ studying geometries of the form Γ(G, F), where G = M11 and Gi is a maximal subgroup of G for all i ∈ I.

For the remainder of this paper G will denote M11. From the [9], the conjugacy classes of the maximal subgroups of G are as follows:

Order Index Mi ∼ 720 11 M1 = M10 ∼ 144 55 M2 = M9 : 2 ∼ 48 165 M3 = M8 : S3 ∼ 120 66 M4 = S5 ∼ 660 12 M5 = L2(11)

For i ∈ {1, ..., 5}, we let Mi denote the conjugacy class of Mi, Mi as given in the S5 previous table. We also set M= i=1 Mi; so M consist of all maximal subgroups of G. Suppose G1 and G2 are maximal subgroups of G with G1 6= G2. Set G12 = G1 ∩ G2. We use Mij(1) to describe {G1,G2,G1 ∩ G2} according to the following scheme: G1 ∈ Mi, G2 ∈ Mj and the ﬁrst case of the intersection of G1 and G2. Mij(2) to describe {G1,G2,G1 ∩ G2} according to the following scheme: G1 ∈ Mi, G2 ∈ Mj and the second case of the intersection of G1 and G2, and so we can deﬁne the other cases similarly. Our notation is as in the [9] with the following addition: Fn a Frobenious group of order n. 6352 Nayil Kilic

3. Rank 2 geometries of M11

In this section, we give, up to conjugacy, all rank 2 geometries for M11 that are firm, residually connected and flag-transitive. Theorem 1. Up to conjugacy in AutG there are 33 firm, residually connected and flag-transitive rank 2 geometries of Γ = Γ(G, {G1,G2}) with G1,G2 ∈ M. These together with the shape and order of G12 are listed in the following table.

Γ G12 | G12 | Γ G12 | G12 | M11(1) M9 72 M12(1) M8 : 2 16 M12(2) M9 72 M13(1) S3 6 M13(2) M8 : 2 16 M14(1) F20 20 M14(2) S4 24 M15(1) A5 60 2 M22(1) 2 4 M22(2) Q8 8 M23(1) C2 2 M23(2) D12 12 M23(3) M8 : 2 16 M24(1) C4 4 M24(2) D8 8 M24(3) D12 12 M25(1) D12 12

M33(1) 1 1 M33(2) C2 2 2 M33(3) 2 4 M33(4) S3 6 M34(1) C2 2 M34(2) S3 6 M34(3) D8 8 M34(4) D12 12 M35(1) S3 6 M35(2) D12 12 2 M44(1) 2 4 M44(2) S3 6 M44(3) D8 8 M45(1) D12 12 M45(2) A5 60

M55(1) A5 60

4. Diagrams for Rank 2 Geometries In this section, we give the diagrams of all rank 2 geometries. All rank 2 primitive geometries of the Mathieu group M11 6353 Γ Diagrams Γ Diagrams Γ Diagrams M11(1) 4.1 M12(1) 4.2 M12(2) 4.3 M13(1) 4.4 M13(2) 4.5 M14(1) 4.6 M14(2) 4.7 M15(1) 4.8 M22(1) 4.9 M22(2) 4.10 M23(1) 4.11 M23(2) 4.12 M23(3) 4.13 M24(1) 4.14 M24(2) 4.15 M24(3) 4.16 M25(1) 4.17 M33(1) 4.18 M33(2) 4.19 M33(3) 4.20 M33(4) 4.21 M34(1) 4.22 M34(2) 4.23 M34(3) 4.24 M34(4) 4.25 M35(1) 4.26 M35(2) 4.27 M44(1) 4.28 M44(2) 4.29 M44(3) 4.30 M45(1) 4.31 M45(2) 4.32 M55(1) 4.33 3 2 3 3 2 3 4.1OO 4.2 OO −1 9 8 44 11 11 11 55 M10 M10 M10 M9 : 2 B = M9 B = M8 : 2

3 3 4 3 2 3 4.3OO 4.4 OO 1 9 7 119 11 55 11 165 M10 M9 : 2 M10 M8 : S3 B = M9 B = S3 Due to Buekenhout(see [3], geometry 2.3).

3 2 4 3 2 3 4.5OO 4.6 OO 2 44 35 5 11 165 66 11 M10 M8 : S3 S5 M10 B = M8 : 2 B = F20

3 2 3 2 2 2 4.7OO 4.8 OO 11 4 11 10 66 11 12 11 S5 M10 L2(11) M10 B = S4 B = A5 6354 Nayil Kilic

Due to Buekenhout(see [3], geometry 2.1).

3 2 3 3 2 3 4.9OO 4.10 OO 35 35 17 17 55 55 55 55 M9 : 2 M9 : 2 M9 : 2 M9 : 2 2 B = 2 B = Q8

3 2 3 4 3 4 4.11OO 4.12 OO 71 23 11 3 165 55 165 55 M8 : S3 M9 : 2 M8 : S3 M9 : 2 B = C2 B = D12

Due to Buekenhout(see [3], geometry 2.6).

6 3 5 3 2 3 4.13OO 4.14 OO 8 2 35 29 165 55 66 55 M8 : S3 M9 : 2 S5 M9 : 2 B = M8 : 2 B = C4 Due to Buekenhout(see [3], geometry 2.5).

3 2 3 4 2 3 4.15OO 4.16 OO 17 14 11 9 66 55 66 55 S5 M9 : 2 S5 M9 : 2 B = D8 B = D12

2 2 2 3 2 3 4.17OO 4.18 OO 11 54 47 47 12 55 165 165 L2(11) M9 : 2 M8 : S3 M8 : S3 B = D12 B = 1 Due to Buekenhout(see [3], geometry 2.2). All rank 2 primitive geometries of the Mathieu group M11 6355

3 2 4 4 2 4 4.19OO 4.20 OO 23 23 11 11 165 165 165 165 M8 : S3 M8 : S3 M8 : S3 M8 : S3 2 B = C2 B = 2

5 3 5 3 2 3 4.21OO 4.22 OO 7 7 23 59 165 165 66 165 M8 : S3 M8 : S3 S5 M8 : S3 B = S3 B = C2 Due to Buekenhout(see [3], geometry 2.8).

4 2 4 4 2 4 4.23OO 4.24 OO 7 19 5 14 66 165 66 165 S5 M8 : S3 S5 M8 : S3 B = S3 B = D8

5 3 5 3 2 3 4.25OO 4.26 OO 3 9 7 109 66 165 12 165 S5 M8 : S3 L2(11) M8 : S3 B = D12 B = S3 Due to Buekenhout(see [3], geometry 2.7).

3 2 4 3 2 3 4.27OO 4.28 OO 3 54 29 29 12 165 66 66 L2(11) M8 : S3 S5 S5 2 B = D12 B = 2 6356 Nayil Kilic

3 2 3 5 2 5 4.29OO 4.30 OO 19 19 14 14 66 66 66 66 S5 S5 S5 S5 B = S3 B = D8

3 2 3 3 3 4 4.31OO 4.32 OO 9 54 1 10 12 66 12 66 L2(11) S5 L2(11) S5 B = D12 B = A5

Due to Buekenhout(see [3], geometry 2.4).

3 2 3 4.33 OO 10 10 12 12 L2(11) L2(11)

B = A5

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Received: August 6, 2014