2 ′ (2, 3, t)-Generations for the Tits 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 ′ T = F4(2) where t is divisor of |T|. Most of the computations were carried out with the aid of computer algebra system GAP [17].

2 ′ 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 of the simple group T, while studying erated by an involution x and an element y of 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 PSL(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 ∈ 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 |T|. 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 | || | | | ∑ 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 |G| i=1 [χi(1G)] [16] and [22]. where χ1, χ2, . . . , χm are the irreducible ∼ 2 ′ 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 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 ∈ 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 ∈ 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) ∈ (C1 × C2 × ... × Ck−1) such d = dim(V/C (2A)) = 32, that g1g2 . . . gk−1 = gk and ⟨g1, g2, . . . , gk−1⟩ ≤ 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 ∈ {8C, 8D, 10A}, 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 ∈ {A, B, C, D} 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 ⟨xi⟩ 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 ∈ {A, B}: 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 ∼ |CG(z)|, z ∈ 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 ∈ {A, B}, follows. ∼ 2F (2)′ 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 ∈ 8C (2, 3, t) 3 In this section we investigate -generations is contained in two conjugate subgroup of each t | | 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 ∈ L3(3) H2 ∼ 32. By considering the maximal subgroups of H = {8, 10, 12, 13, 16}. 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 ∈ 11 2 generated. Thus, {8A, 8B, 8C, 8D, 10A}. ∗ ∆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. = 112 − 4(20) − 32 = 0. Therefore, the Tits simple group T is not Since no maximal subgroup of each H and K is (2B, 3A, 8C)-generated. (2A, 3A, 12Z)-generated, we obtain Case (2B, 3A, 8D): In this case, ∗ ∆T(2B, 3A, 8D) = 112. We prove that Tits ∆T(2A, 3A, 12Z) = ∆T(2A, 3A, 12Z) − ∗ simple group T is not (2B, 3A, 8D)-generated by ΣH (2A, 3B, 12Z) constructing the (2B, 3A, 8D)-generated subgroup − ∗ 4ΣK (2A, 3A, 12Z) of the group He explicitly. We use the ”standard − − generators” of the group T given by Wilson in = 32 12 2(15) < 0 [21]. The group T has a 26-dimensional irreducible and the non-generation of the group Tits by the triple representation over GF(2). Using this representation (2A, 3A, 12Z) follows. we generate the Tits group T = ⟨a, b⟩, where a Next, suppose That X = B. There are six and b are 26 × 26 matrices over GF(2) with orders maximal subgroups of the group T having non-empty 2 and 3 respectively such that ab has order 13. intersection with each conjugacy class in the triple Using GAP, we see that a ∈ 2A, b ∈ 3A. We (2B, 3A, 12Z), are isomorphic to H = L (3):2 produce c = (ababab2)6, p = abababab2abab2ab2, ∼ 1 3 ∼ − (two non-conjugate copies), H = L (25), H = d = (acp)6, x = p16dp 16 such that c, d, x ∈ 2B, 2 2 3 22.[28]:S and H = 52:4A (two non-conjugate p ∈ 10A and xb ∈ 8D. Let H = ⟨x, b⟩ then H < T 3 4 4 ∼ copies). Further, a fixed element of order 12 in with H = L (3):2. Since no maximal subgroup 3 Tits group is contained in a unique conjugate sub- of H is (2B, 3A, 8D)-generated, that is no proper groups of each of H ,H ,H and H . We calculate subgroup of H is (2B, 3A, 8D)-generated and we 1 2 3 4 ∗ ∆ (2B, 3A, 12Z) = 84, Σ (2B, 3A, 12Z) = 27, have Σ (2B, 3A, 8D) = Σ (2B, 3A, 8D). Since T H1 H H Σ (2B, 3A, 12Z) = 24, Σ (2B, 3A, 12Z) = 12 Σ (2B, 3A, 8D) = 28 and z ∈ 8D is contained in H2 H3 H and Σ (2B, 3A, 12Z) = 0. Since no maximal sub- exactly two conjugate subgroups of each copy of H, H4 ∗ group of each of the groups H , H , H and H is we obtain that ∆ (2B, 3A, 8D) = 0. Hence the Tits 1 2 3 4 T (2B, 3A, 12Z)-generated. We conclude that simple group T is not (2B, 3A, 8D)-generated. This completes the lemma. ∗ ∆T(2B, 3A, 12Z) = ∆T(2B, 3A, 12Z) − ∗ Lemma 5 The Tits group T is 2ΣH1 (2B, 3A, 12Z) (2B, 3A, 10A)−generated. − ∗ ΣH2 (2B, 3A, 12Z) −Σ∗ (2B, 3A, 12Z) Proof. Up to isomorphism, the only maximal sub- H3 − − − groups having non-empty intersection with any con- = 84 2(27) 24 12 < 0. jugacy class in the triple (2B, 3A, 10A) are isomor- ∼ ∼ Therefore Tits group T is not (2B, 3A, 12Z)- phic to H = 22.[28]:S , K = A ·22(two non- 3 6 generated. This completes the proof. conjugate copies). Since ∆T (2B, 3A, 10A) = 100 and Σ (2B, 3A, 10A) = 0 = Σ (2B, 3A, 10A). H K Lemma 7 The Tits group T is (2X, 3A, 13Z)- we conclude that no maximal subgroup of T is generated where X,Z ∈ {A, B} if and only if X = A (2B, 3A, 10A)-generated. Thus ∗ Proof. First we consider the case X = A. The struc- ∆T(2B, 3A, 10A) = ∆T(2B, 3A, 10A) = 100 ture constant ∆T(2A, 3A, 13Z) = 13. The fusion and the (2B, 3A, 10A)-generation of Tits group T fol- maps of the maximal subgroup of Tits group T into lows. the group T shows that there is no maximal subgroup of T has non-empty intersection with the classes in the Lemma 6 The Tits group T is not triple (2A, 3A, 13Z). That is no maximal subgroup of (2X, 3A, 12Z)−generated where X,Z ∈ {A, B}. T is (2A, 3A, 13Z)-generated. Hence,

∗ Proof. First we consider the case X = A. ∆T(2A, 3A, 13Z) = ∆T(2A, 3A, 13Z) = 13 > 0 The maximal subgroups of the group T that may contain (2A, 3A, 12Z)-generated subgroups are iso- which implies that the Tits group T is (2A, 3A, 13Z)- ∼ 2 8 ∼ 2 morphic to H = 2 .[2 ]:S3 and K = 5 :4A4 generated for Z ∈ {A, B}. (two non-conjugate copies). We compute that Next suppose that X = B. Up to iso- ∆T (2A, 3A, 12Z) = 32, ΣH (2A, 3A, 12Z) = 12 morphism, the only maximal subgroups of T hav- and ΣK (2A, 3A, 12Z) = 15. A fixed element of ing non-empty intersection with each conjugacy order 12 in T is contained in a unique conjugate class in the triple (2B, 3A, 13Z) are isomorphic subgroup of H and two conjugate subgroups of K. to L3(3):2 (two non-conjugate copies) and L2(25). Further a fixed element of order 13 in the Tits Proof. This is merely a restatement of the lemmas in group T is contained in a unique conjugate of each this section. of L3(3):2 and in three conjugate of L2(25) sub- groups. We compute that ∆T (2B, 3A, 13Z) = 104, Acknowledgements: This research was supported by ΣL (3):2(2B, 3A, 13Z) = ΣL (3)(2B, 3A, 13A) = the King Khalid University, Saudi Arabia under re- 3 3 search grant No. 244-44. 13 and ΣL2(25)(2B, 3A, 13Z) = 26. Now by con- sidering the maximal subgroups of L3(3) and L2(25), we see that no maximal subgroup of the groups References: L3(3) and L2(25) is (2B, 2A, 13Z)-generated. It fol- lows that no proper subgroup of L3(3) or L2(25) is [1] F. Ali, (2, 3, p)-Generations for the Fischer (2B, 3A, 13Z)-generated. Thus we have Group F i23, submitted. [2] F. Ali and M. A. F. Ibrahim, On the ranks of Con- ∆∗ (2B, 3A, 13Z) = ∆ (2B, 3A, 13Z) T T way groups Co2 and Co3, J. Algebra Appl., 4, −2Σ∗ (2B, 3A, 13Z) 2005, pp. 557–565. L3(3) −3Σ∗ (2B, 3A, 13Z) [3] F. Ali and M. A. F. Ibrahim, On the ranks of Con- L2(25) way group Co , Proc. Japan Acad., 81A, 2005, − − − 1 = 104 2(13) 3(26) 12 = 0, pp. 95–98. proving non-generation of the Tits group T by the [4] F. Ali and M. A. F. Ibrahim, On the ranks of triple (2B, 3A, 13Z), where Z ∈ {A, B}. HS and McL, Utilitas Mathematica, to appear in Volume 70, July 2006. Lemma 8 The Tits group T is (2X, 3A, 16Z)- [5] F. Ali and M. A. F. Ibrahim, (2, 3, t)-Generations generated, where X ∈ {A, B} and Z ∈ for the Held’s He, submitted. {A, B, C, D}. [6] M. D. E. Conder, Hurwitz groups: A brief sur- vey, Bull. Amer. Math. Soc., 23, 1990, pp. 359– Proof. We treat two cases separately. 370. Case (2A,3A,16Z): The structure constant [7] M. D. E. Conder, Some results on quotients of ∆T(2A, 3A, 16Z) = 16. We observe that the triangle groups, Bull. Australian Math. Soc., 30, 2 8 group isomorphic to 2 .[2 ]:S3 is the only maximal 1984, pp. 73–90. subgroup of T that may contain (2A, 3A, 16Z)- [8] M. D. E. Conder, R. A. Wilson and A. J. Woldar, generated subgroups. However we calculate The symmetric genus of sporadic groups, ∼ 2 8 ΣH (2A, 3A, 16Z) = 0 for H = 2 .[2 ]:S3 and hence ∗ Proc. Amer. Math. Soc., 116, 1992, pp. 653–663. ∆T (2A, 3A, 16Z) = ∆T (2A, 3A, 16Z) = 16 > 0, [9] J. H. Conway, R. T. Curtis, S. P. Norton, proving that (2A, 3A, 16Z) is a generating triple of R. A. Parker, R. A. Wilson, ATLAS of Finite the Tits group. ∼ Groups, Oxford Univ. Press (Clarendon), Ox- Case (2B,3A,16Z): Up to isomorphism, H = ford, 1985. 22.[28]:S is the only one maximal subgroup of T that 3 [10] L. Di Martino and C. Tamburini, 2-Generation may admit (2B, 3A, 16Z)-generated subgroups. A of finite simple groups and some related topics, fixed element of order 16 in the Tits group T is con- Generators ans Relations in Groups and Geom- tained in a unique conjugate subgroups of H. Since etry, A. Barlotti et al., Kluwer Acad. Publ., New ∆ (2B, 3A, 16Z) = 112, Σ (2B, 3A, 16Z) = 32, T H York (1991), pp. 195–233. we conclude that [11] S. Ganief and J. Moori, (2, 3, t)-Generations for ∗ ≥ − the Janko group J , Comm. Algebra, 23, 1995, ∆T(2B, 3A, 16Z) 112 32 = 80 > 0 3 pp. 4427–4437. and the (2B, 3A, 16Z)-generation of T follows. [12] M. A. F. Ibrahim, On the ranks of certain spo- radic simple groups by Suzuki, Thompson and We now summarize our results in the next theo- Rudvalis, Algebras, Groups and Geometries, to rem. appear. [13] M. A. F. Ibrahim, On the ranks of Harada-Norton Theorem 9 Let tX be a conjugacy class of the sporadic group HN, Int. Math. J., to appear. Tits simple group T. The group T is (2A, 3A, tX)- [14] I. M. Isaacs, Character Theory of Finite Groups, generated if and only if tX ∈ {13Y, 16Z} where Dover, New-York, 1976. Y ∈ {A, B} and Z ∈ {A, B, C, D}. Further, [15] J. Moori, (2, 3, p)-Generations for the Fischer the group T is (2B, 3A, tX)-generated if and only if group F , Comm. Algebra, 22, 1994, pp. 4597– tX ∈ {8Y, 10A, 16Z}. 22 4610. [16] J. Moori, On the ranks of the Fischer group F22, Math. Japonica, 43, 1996, pp. 365–367. [17] The GAP Group, GAP-Groups, Al- gorithms and Programming, Version 4.3 , Aachen, St Andrews, 2002, (http://www.gap-system.org). [18] L. L. Scott, Matrices and Cohomology, Ann. of Math. , 105, 1977, pp. 473–492. [19] K. B. Tchakerian, The maximal subgroups of the Tits simple group, C. R. Acad. Bulgare Sci., 34, 1981, pp. 16–37. [20] R. A. Wilson, The geometry and maximal sub- groups of the simple groups of A. Rudvalis and J. Tits, Proc. London Math. Soc. , 48, 1984, pp. 553–563. [21] R. A. Wilson et al., A world-wide- web Atlas of Group Representations, (http://web.mat.bham.ac.uk/atlas). [22] A. J. Woldar, On Hurwitz generation and genus actions of sporadic groups, Illinois Math. J., 33, 1989, pp. 416–437. [23] A. J. Woldar, Sporadic simple groups which are Hurwitz, J. Algebra, 144, 1991, pp. 443–450. [24] A. J. Woldar, Representing M11, M12, M22 and M23 on surfaces of least genus, Comm. Algebra, 18, 1990, pp. 15–86.