Proceedings of the 9th WSEAS International Conference on Applied Mathematics, Istanbul, Turkey, May 27-29, 2006 (pp362-366)

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 m f

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 for 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 F . In [11], Ganief 22 and taken mainly from [1], [2], [3], [4], [5], [15] and Moori established (2, 3, t)-generations of the third and [11]. In particular, for a finite group with Janko group J . In a series of papers [1], [2], [3], G 3 conjugacy classes of its elements [4], [5], [12] and [13], the authors studied (2, 3)- C1,C2,...,Ck generation and generation by conjugate elements of and gk a fixed representative of Ck, we denote the number of dis- the sporadic simple groups Co , Co , Co , He, HN, ∆(G) = ∆G(C1,C2,...,Ck) 1 2 3 tinct tuples with such Suz, Ru, HS, McL, T h and F i . The present arti- (g1, g2, . . . , gk−1) gi ∈ Ci 23 that . It is well known that cle is devoted to the study of (2, 3, t)-generations for g1g2 . . . gk−1 = gk is structure constant for the con- the Tits simple group T, where t is any divisor of |T|. ∆G(C1,C2,...,Ck) jugacy classes and can easily be com- For more information regarding the study of (2, 3, t)- C1,C2,...,Ck generations, generation by conjugate elements as well puted from the character table of G (see [14], p.45) as computational techniques used in this article, the by the following formula ∆G(C1,C2,...,Ck) = |C1||C2|...|Ck−1| m χi(g1)χi(g2)...χi(gk−1)χi(gk) × P k−2 reader is referred to [1], [2], [3], [4], [5], [11], [15], |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. Proceedings of the 9th WSEAS International Conference on Applied Mathematics, Istanbul, Turkey, May 27-29, 2006 (pp362-366)

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 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 of over ∆H (c1, c2, . . . , ck) H 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 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 ∈ {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) = ∆G(lX, mY, nZ) < ing non-empty intersection with any conjugacy class ∗ ∼ |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. The Tits group T ∼ 2 ′ has conjugacy classes = F4(2) 8 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 L3(3):2 (two non-conjugate copies) and H2 =∼ lutions namely 2A and 2B. 2 8 2 .[2 ]:S3. Further, a fixed element z ∈ 8C In this section we investigate (2, 3, t)-generations is contained in two conjugate subgroup of each for the Tits group T where is a divisor of T . It is a t | | copy of H and in a unique conjugate subgroup well known fact that if a group is -generated 1 G (2, 3, t) of H . A simple computation using GAP reveals simple group, then (see [7] for 2 1/2 + 1/3 + 1/t < 1 that ∆ (2B, 3A, 8C) = 112, Σ (2B, 3A, 8C) = details). It follows that for the -generations of T H1 (2, 3, t) 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 ∗ structure constants ∆T(2A, 3A, 8A) = 0 and −ΣH2 (2B, 3A, 8C) ∆T(2A, 3A, 8B) = 0. = 112 − 4(20) − 32 = 0. Proceedings of the 9th WSEAS International Conference on Applied Mathematics, Istanbul, Turkey, May 27-29, 2006 (pp362-366)

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 −4Σ∗ (2A, 3A, 12Z) of the group He explicitly. We use the ”standard K 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 = ha, bi, 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 = hx, bi 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). Proceedings of the 9th WSEAS International Conference on Applied Mathematics, Istanbul, Turkey, May 27-29, 2006 (pp362-366)

Further a fixed element of order 13 in the Tits (ii) (2B, 3A, tX)-generated if and only if tX ∈ group T is contained in a unique conjugate of each {8Y, 10A, 16Z}. of L3(3):2 and in three conjugate of L2(25) sub- groups. We compute that ∆ (2B, 3A, 13Z) = 104, T Acknowledgements: This research was supported by ΣL3(3):2(2B, 3A, 13Z) = ΣL3(3)(2B, 3A, 13A) = 13 and Σ (2B, 3A, 13Z) = 26. Now by con- the King Khalid University, Saudi Arabia under re- L2(25) search grant No. 244-44. sidering the maximal subgroups of L3(3) and L2(25), we see that no maximal subgroup of the groups L (3) and L (25) is (2B, 2A, 13Z)-generated. It fol- 3 2 References: lows that no proper subgroup of L3(3) or L2(25) is (2B, 3A, 13Z)-generated. Thus we have [1] F. Ali, (2, 3, p)-Generations for the Fischer Group F i23, submitted. ∆∗ (2B, 3A, 13Z) = ∆ (2B, 3A, 13Z) T T [2] F. Ali and M. A. F. Ibrahim, On the ranks of Con- −2Σ∗ (2B, 3A, 13Z) L3(3) way groups Co2 and Co3, J. Algebra Appl., 4, ∗ 2005, pp. 557–565. −3ΣL2(25)(2B, 3A, 13Z) = 104 − 2(13) − 3(26) − 12 = 0, [3] F. Ali and M. A. F. Ibrahim, On the ranks of Con- way group Co1, Proc. Japan Acad., 81A, 2005, proving non-generation of the Tits group T by the pp. 95–98. triple (2B, 3A, 13Z), where Z ∈ {A, B}. [4] F. Ali and M. A. F. Ibrahim, On the ranks of HS and McL, Utilitas Mathematica, to appear Lemma 8 The Tits group T is (2X, 3A, 16Z)- in Volume 70, July 2006. generated, where X ∈ {A, B} and Z ∈ {A, B, C, D}. [5] F. Ali and M. A. F. Ibrahim, (2, 3, t)-Generations for the Held’s He, submitted. Proof. We treat two cases separately. [6] M. D. E. Conder, Hurwitz groups: A brief sur- Case (2A,3A,16Z): The structure constant vey, Bull. Amer. Math. Soc., 23, 1990, pp. 359– ∆T(2A, 3A, 16Z) = 16. We observe that the 370. 2 8 group isomorphic to 2 .[2 ]:S3 is the only maximal [7] M. D. E. Conder, Some results on quotients of subgroup of T that may contain (2A, 3A, 16Z)- triangle groups, Bull. Australian Math. Soc., 30, generated subgroups. However we calculate 2 8 1984, pp. 73–90. ΣH (2A, 3A, 16Z) = 0 for H =∼ 2 .[2 ]:S3 and hence ∗ [8] M. D. E. Conder, R. A. Wilson and A. J. Woldar, ∆T (2A, 3A, 16Z) = ∆T (2A, 3A, 16Z) = 16 > 0, proving that (2A, 3A, 16Z) is a generating triple of The symmetric genus of sporadic groups, the Tits group. Proc. Amer. Math. Soc., 116, 1992, pp. 653–663. Case (2B,3A,16Z): Up to isomorphism, H =∼ [9] J. H. Conway, R. T. Curtis, S. P. Norton, 2 8 ATLAS 2 .[2 ]:S3 is the only one maximal subgroup of T that R. A. Parker, R. A. Wilson, of Finite may admit (2B, 3A, 16Z)-generated subgroups. A Groups, Oxford Univ. Press (Clarendon), Ox- fixed element of order 16 in the Tits group T is con- ford, 1985. tained in a unique conjugate subgroups of H. Since [10] L. Di Martino and C. Tamburini, 2-Generation ∆T (2B, 3A, 16Z) = 112, ΣH (2B, 3A, 16Z) = 32, of finite simple groups and some related topics, we conclude that Generators ans Relations in Groups and Geom- ∗ etry, A. Barlotti et al., Kluwer Acad. Publ., New ∆T(2B, 3A, 16Z) ≥ 112 − 32 = 80 > 0 York (1991), pp. 195–233. and the (2B, 3A, 16Z)-generation of T follows. [11] S. Ganief and J. Moori, (2, 3, t)-Generations for the Janko group J3, Comm. Algebra, 23, 1995, pp. 4427–4437. [12] M. A. F. Ibrahim, On the ranks of certain spo- 4 Conclusion radic simple groups by Suzuki, Thompson and Rudvalis, Algebras, Groups and Geometries, to Let tX be a conjugacy class of the Tits simple group appear. T. Then Tits simple group T is [13] M. A. F. Ibrahim, On the ranks of Harada-Norton (i) (2A, 3A, tX)-generated if and only if tX ∈ sporadic group HN, Int. Math. J., to appear. {13Y, 16Z} where Y ∈ {A, B} and Z ∈ [14] I. M. Isaacs, Character Theory of Finite Groups, {A, B, C, D}, Dover, New-York, 1976. Proceedings of the 9th WSEAS International Conference on Applied Mathematics, Istanbul, Turkey, May 27-29, 2006 (pp362-366)

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