2 0 (2; 3; t)-Generations for the Tits simple group F4(2) MOHAMMED ALI FAYA IBRAHIM FARYAD ALI King Khalid University King Khalid University Department of Mathematics Department of Mathematics P.O. Box 9004, Abha P.O. Box 9004, Abha SAUDI ARABIA SAUDI ARABIA mafi[email protected] [email protected] Abstract: A group G is said to be (2; 3; t)-generated if it can be generated by two elements x and y such that o(x) = 2, o(y) = 3 and o(xy) = t. In this paper, we determine (2; 3; t)-generations of the Tits simple group ∼ 2 0 T = F4(2) where t is divisor of jTj. Most of the computations were carried out with the aid of computer algebra system GAP [17]. 2 0 Key–Words: Tits group F4(2) , simple group, (2; 3; t)-generation, generator. 1 Introduction was first determined by Tchakerian [19]. Later but independently, Wilson [20] also determined the max- A group G is called (2; 3; t)-generated if it can be gen- imal subgroups of the simple group T, while studying erated by an involution x and an element y of order 3 the geometry of the simple groups of Tits and Rud- such that o(xy) = t. The (2; 3)-generation problem valis. has attracted a vide attention of group theorists. One For basic properties of the Tits group T and infor- reason is that (2; 3)-generated groups are homomor- mation on its subgroups the reader is referred to [20], phic images of the modular group P SL(2; Z), which [19]. The ATLAS of Finite Groups [9] is an impor- is the free product of two cyclic groups of order two tant reference and we adopt its notation for subgroups, and three. The motivation of (2; 3)-generation of sim- conjugacy classes, etc. Computations were carried out ple groups also came from the calculation of the genus with the aid of GAP [17]. of finite simple groups [22]. The problem of finding the genus of finite simple group can be reduced to one of generations (see [24] for details). 2 Preliminary Results Moori in [15] determined the (2; 3; p)-generations Throughout this paper our notation is standard of the smallest Fischer group F22. In [11], Ganief and Moori established (2; 3; t)-generations of the third and taken mainly from [1], [2], [3], [4], [5], [15] G Janko group J . In a series of papers [1], [2], [3], and [11]. In particular, for a finite group with 3 C ;C ;:::;C [4], [5], [12] and [13], the authors studied (2; 3)- 1 2 k conjugacy classes of its elements g C generation and generation by conjugate elements of and k a fixed representative of k, we denote ∆(G) = ∆G(C1;C2;:::;Ck) the number of dis- the sporadic simple groups Co1; Co2; Co3, He, HN, tinct tuples (g1; g2; : : : ; gk−1) with gi 2 Ci such Suz, Ru, HS, McL, T h and F i23. The present arti- g g : : : g − = g cle is devoted to the study of (2; 3; t)-generations for that 1 2 k 1 k. It is well known that ∆ (C ;C ;:::;C ) the Tits simple group T, where t is any divisor of jTj. G 1 2 k is structure constant for the con- C ;C ;:::;C For more information regarding the study of (2; 3; t)- jugacy classes 1 2 k and can easily be com- G 45 generations, generation by conjugate elements as well puted from the character table of (see [14], p. ) by the following formula ∆G(C1;C2;:::;Ck) = as computational techniques used in this article, the j jj j j j P C1 C2 ::: Ck−1 × m χi(g1)χi(g2)...χi(gk−1)χi(gk) reader is referred to [1], [2], [3], [4], [5], [11], [15], k−2 jGj i=1 [χi(1G)] [16] and [22]. where χ1; χ2; : : : ; χm are the irreducible ∼ 2 0 The Tits group T = F4(2) is a simple group complex characters of G. Further, let 11 3 2 ∗ ∗ of order 17971200 = 2 :3 :5 :13. The group T is ∆ (G) = ∆G(C1;C2;:::;Ck) denote the num- a subgroup of the Rudvalis sporadic simple group Ru ber of distinct tuples (g1; g2; : : : ; gk−1) with of index 8120. The group T also sits maximally inside gi 2 Ci and g1g2 : : : gk−1 = gk such that G =< ∗ the smallest Fischer group F i22 with index 3592512. g1; g2; : : : ; gk−1 >. If ∆G(C1;C2;:::;Ck) > 0, The maximal subgroups of the Tits simple group T then we say that G is (C1;C2;:::;Ck)-generated. If H is any subgroup of G containing the fixed ele- The group T acts on 78-dimensional irreducible ment gk 2 Ck, then ΣH (C1;C2;:::;Ck−1;Ck) complex module V . We apply Scott’s theorem (cf. denotes the number of distinct tuples Theorem 1) to the module V and compute that (g1; g2; : : : ; gk−1) 2 (C1 × C2 × ::: × Ck−1) such d = dim(V=C (2A)) = 32; that g1g2 : : : gk−1 = gk and hg1; g2; : : : ; gk−1i ≤ H 2A V where ΣH (C1;C2;:::;Ck) is obtained by summing d3A = dim(C=CV (3A)) = 54 the structure constants ∆H (c1; c2; : : : ; ck) of H over d8C = dim(V=CV (8C)) = 68; all H-conjugacy classes c1; c2; : : : ; ck−1 satisfying d8D = dim(V=CV (8D)) = 68 ci ⊆ H \ Ci for 1 ≤ i ≤ k − 1. For the description of the conjugacy classes, the d10A = dim(V=CV (10A)) = 68 character tables, permutation characters and informa- Now, if the group T is (2A; 3A; tX)-generated, where tion on the maximal subgroups readers are referred to tX 2 f8C; 8D; 10Ag, then by Scott’s theorem we ATLAS [9]. A general conjugacy class of elements must have of order n in G is denoted by nX. For example 2A represents the first conjugacy class of involutions in a d2A + d3A + dtX ≥ 2 × 78 = 156: group G. The following results in certain situations are very However, d2A +d3A +dtX = 154, and non-generation effective at establishing non-generations. of the group T by these triples follows. Theorem 1 (Scott’s Theorem, [8] and [18]) Let Lemma 4 The Tits simple group T is x1; x2; : : : ; xm be elements generating a group G with (2B; 3A; 8Z)−generated, where Z 2 fA; B; C; Dg x1x2···xn = 1G, and V be an irreducible module for if and only if Z = A or B. G of dimension n ≥ 2. Let CV (xi) denote the fixed point space of hxii on V , and let di is the codimension Proof. Our main proof will consider the following of V=CV (xi). Then d1 + d2 + ··· + dm ≥ 2n . three cases. Case (2B; 3A; 8Z), where Z 2 fA; Bg: We Lemma 2 ([8]) Let G be a finite centerless compute ∆T (2B; 3A; 8Z) = 128. Amongst the max- group and suppose lX, mY , nZ are G-conjugacy imal subgroup of T, the only maximal subgroups hav- classes for which ∆∗(G) = ∆∗ (lX; mY; nZ) < ing non-empty intersection with any conjugacy class ∗ G ∼ jCG(z)j; z 2 nZ. Then ∆ (G) = 0 and therefore in the triple (2B; 3A; tZ) is isomorphic to H = 2 8 G is not (lX; mY; nZ)-generated. 2 :[2 ]:S3. However ΣH (2B; 3A; 8Z) = 0, which means that H is not (2B; 3A; 8Z)-generated. Thus ∗ ∆T(2B; 3A; 8Z) = ∆T (2B; 3A; 8Z) = 128 > 3 (2; 3; t)-Generations of Tits group 0, and the (2B; 3A; 8Z)-generation of T, for Z 2 fA; Bg, follows. ∼ 2F (2)0 8 The Tits group T = 4 has conjugacy classes Case (2B; 3A; 8C): The only maximal sub- of its maximal subgroups as determined by Wilson groups of the group T that may contain (2B; 3A; 8C)- [20] and listed in the ATLAS [9]. The group T has ∼ generated subgroups, up to isomorphism, are H = 22 conjugacy classes of its elements including 2 invo- 1 ∼ L (3):2 (two non-conjugate copies) and H = 2A 2B 3 2 lutions namely and . 22:[28]:S . Further, a fixed element z 2 8C (2; 3; t) 3 In this section we investigate -generations is contained in two conjugate subgroup of each t j j for the Tits group T where is a divisor of T . It is a copy of H and in a unique conjugate subgroup G (2; 3; t) 1 well known fact that if a group is -generated of H . A simple computation using GAP reveals 1=2 + 1=3 + 1=t < 1 2 simple group, then (see [7] for that ∆ (2B; 3A; 8C) = 112, Σ (2B; 3A; 8C) = (2; 3; t) T H1 details). It follows that for the -generations of Σ (2B; 3A; 8C) = 20 and Σ (2B; 3A; 8C) = the Tits simple group T, we only need to consider t 2 L3(3) H2 ∼ 32. By considering the maximal subgroups of H = f8; 10; 12; 13; 16g. 11 L3(3) and H2, we see that no maximal subgroup of H and H is (2B,3A,8C)-generated and hence no Lemma 3 The Tits simple group T is not 11 2 proper subgroup of H and H is (2B; 3A; 8C)- (2A; 3A; tX)-generated for any tX 2 11 2 generated. Thus, f8A; 8B; 8C; 8D; 10Ag. ∗ ∆T(2B; 3A; 8C) = ∆T(2B; 3A; 8C) Proof. For the triples (2A; 3A; 8A) and (2A; 3A; 8B) ∗ −4Σ (2B; 3A; 8S) non-generation follows immediately since the H11 −Σ∗ (2B; 3A; 8C) structure constants ∆T(2A; 3A; 8A) = 0 and H2 ∆T(2A; 3A; 8B) = 0.