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On the ranks of the Conway groups ∗ Co2 and Co3 Faryad Ali y and Mohammed Ali Faya Ibrahim z Department of Mathematics, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia. Abstract Let G be a finite group and X a conjugacy class of G. We denote rank(G:X) to be the minimum number of elements of X generating G. In the present article, we determine the ranks of the Conway groups Co2 and Co3. Computations were carried with the aid of computer algebra system GAP [18]. 1 Introduction and Preliminaries There has recently been some interest in generation of simple groups by their conju- gate involutions. It is well known that sporadic simple groups are generated by three conjugate involutions (see [5]). If a group G = ha; bi is perfect and a2 = b3 = 1 then clearly G is generated by three conjugate involutions a, ab and ab2 (see [6]). Moori [15] proved that the Fischer group F i22 can be generated by three conjugate involutions. The work of Liebeck and Shalev [12] shows that all but finitely many classical groups can be generated by three involutions. However, the problem of finding simple clas- sical groups which can be generated by three conjugate involutions is still very much open. The generation of a simple group by its conjugate elements in this context is of some interest. Therefore, we concentrate on the generation of simple groups by their conjugate elements and investigate two sporadic simple groups. Suppose that G is a finite group and X ⊆ G. We denote the rank of X in G by rank(G:X), the minimum number of elements of X generating G. This paper focuses ∗Dedicated to Professor Jamshid Moori on the occasion of his 60th birthday yE-Mail: [email protected] zE-Mail: mafi[email protected] 1 1 INTRODUCTION AND PRELIMINARIES 2 on the determination of rank(G:X) where X is a conjugacy class of G and G is a sporadic simple group. Moori in [13], [14] and [15] proved that rank(F i22:2A) 2 f5; 6g and rank(F i22:2B) = rank(F i22:2C) = 3 where 2A; 2B and 2C are the conjugacy classes of involutions of the smallest Fischer group F i22 as presented in the ATLAS [3]. The work of Hall and Soicher [9] show that rank(F i22:2A) = 6. Moori [16] determined the ranks of the Janko groups J1;J2 and J3. More recently, in [1], the authors investigated the ranks of Higman-Sims group HS and McLaughlin group McL: In the present article we continue our study on the ranks of sporadic simple groups and determine the ranks of the Conway's sporadic simple groups Co2 and Co3. For basic properties of Co2 and Co3, character tables of these groups and their maximal subgroups etc. we use ATLAS [3] and [18]. For detailed information about the computational techniques used in this paper the reader is encouraged to consult [1], [8], [15], and [16]. Next we discuss some background material and introduce the notation. We adopt the same notation as in the above mentioned papers. In particular, if G is a finite group, C1;C2; ··· ;Ck are the conjugacy classes of its elements and gk is a fixed representative of Ck, then ∆G(C1;C2; ··· ;Ck) denotes the number of distinct tuples (g1; g2; ··· ; gk−1) 2 (C1 × C2 × · · · × Ck−1) such that g1g2 ··· gk−1 = gk. It is well known that ∆G(C1;C2; ··· ;Ck) is the structure constant of G for the conjugacy classes C1;C2; ··· ;Ck and can be computed from the character table of G (see [11], p.45 ) by the following formula m jC1jjC2j · · · jCk−1j X χi(g1)χi(g2) ··· χi(gk−1)χi(gk) ∆ (C ;C ; ··· ;C ) = × G 1 2 k jGj [χ (1 )]k−2 i=1 i G ∗ where χ1; χ2; ··· ; χm are the irreducible complex characters of G. Also, ∆G(C1;C2; ··· ;Ck) denotes the number of distinct tuples (g1; g2; ··· ; gk−1) 2 (C1 × C2 × · · · × ∗ Ck−1) such that g1g2 ··· gk−1 = gk and G = hg1; g2; ··· ; gk−1i. If ∆G(C1;C2; ··· ;Ck) > 0, then we say that G is (C1;C2; ··· ;Ck)-generated. If H any subgroup of G contain- ing the fixed element gk 2 Ck, then ΣH (C1;C2; ··· ;Ck−1;Ck) denotes the number of distinct tuples (g1; g2; ··· ; gk−1) 2 (C1 × C2 × · · · × Ck−1) such that g1g2 ··· gk−1 = gk and hg1; g2; ··· ; gk−1i ≤ H where ΣH (C1;C2; ··· ;Ck) is obtained by summing the structure constants ∆H (c1; c2; ··· ; ck) of H over all H-conjugacy classes c1; c2; ··· ; ck−1 satisfying ci ⊆ H \ Ci for 1 ≤ i ≤ k − 1. The ATLAS serves as a valuable source of information and we use the Atlas notation for conjugacy classes, maximal subgroups, character tables, permutation characters, etc. A general conjugacy class of elements of order n in G is denoted by nX. For examples, 2A represents the first conjugacy class of involutions in a group G. We will use the maximal subgroups and the permutations characters of Co2 and Co3 on the conjugates (right cosets ) of the maximal subgroups listed in the ATLAS [3] extensively. 2 RANKS OF CO2 3 The following results will be crucial in determining the ranks of finite groups. Lemma 1 (Moori [16]) Let G be a finite simple group such that G is (lX; mY; nZ)- generated. Then G is (lX; lX; ··· ; lX; (nZ)m)-generated. | {z } m−times Corollary 2 Let G be a finite simple group such that G is (lX; mY; nZ)-generated, then rank(G : lX) ≤ m. Proof: Immediately follows from Lemma 1. Lemma 3 (Conder et al. [4]) Let G be a simple (2X; mY; nZ)-generated group. Then G is (mY; mY; (nZ)2)-generated. 2 Ranks of Co2 18 6 3 The Conway group Co2 is a sporadic simple group of order 2 :3 :5 :7:11:23 with 11 conjugacy classes of maximal subgroups. It has 60 conjugacy classes of its elements including three conjugacy classes of involutions, namely 2A; 2B and 2C. The group Co2 acts primitively on a set Ω of 2300 points. The point stabilizer of this action is isomorphic to U6(2):2 and the orbits have length 1, 891 and 1408. The permutation character of Co2 on the cosets of U6(2):2 is given by χU6(2):2 = 1a + 275a + 2024a For basic properties of Co2 and computational techniques, the reader is encouraged to consult [1], [2], [8] and [19]. We now compute the rank of each conjugacy class of Co2. It is well known that every sporadic simple group can be generated by three involutions (see [6]). In the following lemmas we prove that Co2 can be generated by three involutions a; b; c 2 2X where X 2 fA; Bg. Lemma 4 Co2 is (2B; 2B; 2B; 23A)-generated. Proof: Simple computation show that the structure constant 4Co2 (2B; 2B; 2B; 23A) = 12696: If z is a fixed element of order 23 in Co2 then there are 12696 distinct triples (α; β; γ) such fα; β; γg ⊂ 2A and αβγ = z: We observe that the only maximal sub- group of Co2 which has order divisible by 23 is M23 and z is contained in a unique conjugate of M23. Hence, any proper (2B; 2B; 2B; 23A)-subgroup of Co2 must be in M23. Furthermore, the 2A-class is the only class which fuses to 2B-class of Co2 in M23. It then follows that ΣM23 (2B; 2B; 2B; 23A) = 3174: Thus the total contribution from M23 to the distinct triples (α; β; γ) with fα; β; γg ⊂ 2A and αβγ = z is equal to 3174. Thus we have ∗ ∆Co2 (2B; 2B; 2B; 23A) ≥ ∆Co2 (2B; 2B; 2B; 23A) − ΣM23 (2B; 2B; 2B; 23A) = 12696 − 3174 > 0: 2 RANKS OF CO2 4 Hence Co2 is (2B; 2B; 2B; 23A)-generated. Lemma 5 rank(Co2 : 2X) = 3 where X 2 fB; Cg. Proof: Let X 2 fB; Cg. Ganief and Moori have shown in [8] that Co2 is (2C; 3A; 23A)- generated. Thus, from the previous lemma and the above stated result from [8] together with application of Corollary 2 imply that rank(Co2 : 2X) ≤ 3. But the case rank(Co2 : 2X) = 2 is not possible since if there are x; y 2 2X such ∼ that Co2 =< x; y >, then Co2 = D2n where n = o(xy). This concludes that rank(Co2 : 2X) = 3 whenever X 2 fB; Cg. Lemma 6 The group Co2 is not (2A; 2A; 2A; tX)-generated for any conjugacy class tX in Co2. Proof: The group Co2 acts on a 275-dimensional irreducible complex module V . Let dnX = dim(V=CV (nX)), the co-dimension of the fix space (in V ) of a representative in nX. Using the character table of Co2 we list in Table I, the values of dnX , for the conjugacy classe nX. TABLE I The co-dimensions dnX = dim(V=CV (nX)) d2A d2B d2C d4B d4C d4D d4E d4F d4G 112 120 132 190 194 196 194 196 204 Set T = f2A; 2B; 2C; 4B; 4C; 4D; 4E; 4F; 4Gg. If the group Co2 is (2A; 2A; 2A; tX)- generated, then by Scott's theorem (see [4] and [17]) we must have d2A + d2A + d2A + dtX ≥ 2 × 275: However, it is clear from Table I that 3 × d2A + dtX < 550 for each tX 2 T and therefore Co2 is not (2A; 2A; 2A; tX)-generated, for each tX 2 T . Next suppose that tX2 = T then computing the structure constants, we see that ∆Co2 (2A; 2A; 2A; tX) < jCCo2 (tX)j for each tX2 = T except for tX = 12H.
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  • The Leech Lattice Λ and the Conway Group ·O Revisited

    The Leech Lattice Λ and the Conway Group ·O Revisited

    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 362, Number 3, March 2010, Pages 1351–1369 S 0002-9947(09)04726-6 Article electronically published on October 20, 2009 THE LEECH LATTICE Λ AND THE CONWAY GROUP ·O REVISITED JOHN N. BRAY AND ROBERT T. CURTIS Dedicated to John Horton Conway as he approaches his seventieth birthday. Abstract. We give a new, concise definition of the Conway group ·Oasfol- lows. The Mathieu group M24 acts quintuply transitively on 24 letters and so acts transitively (but imprimitively) on the set of 24 tetrads. We use this 4 24 action to define a progenitor P of shape 2 4 :M24; that is, a free product of cyclic groups of order 2 extended by a group of permutations of the involutory generators. A simple lemma leads us directly to an easily described, short relator, and factoring P by this relator results in ·O. Consideration of the lowest dimension in which ·O can act faithfully produces Conway’s elements ξT and the 24–dimensional real, orthogonal representation. The Leech lattice is obtained as the set of images under ·O of the integral vectors in R24. 1. Introduction The Leech lattice Λ was discovered by Leech [14] in 1965 in connection with the packing of non-overlapping identical spheres into the 24–dimensional space R24 so that their centres lie at the lattice points; see Conway and Sloane [9]. Its con- struction relies heavily on the rich combinatorial structure underlying the Mathieu group M24. The full group of symmeries of Λ is, of course, infinite, as it contains all translations by a lattice vector.