Some exceptional Beauville structures more

Some Exceptional Beauville Structures Ben Fairbairn Departmento de Matematicas, Universidad de los Andes, Carrera 1, No 18A-12, Bogot´ , Colombia a bt.fairbairn20@uniandes.edu.co 2000 Mathematics subject classification: 14J29, 30F10, 20D06, 20D08 Keywords: Beauville structure, Beauville surface, Complex surface, quasisimple group Abstract We first show that every quasisimple sporadic group possesses an unmixed strongly real Beauville structure aside from the Mathieu groups M11 and M23 (and possibly 2˙B and M). We go on to show that no almost simple sporadic group possesses a mixed Beauville structure. We go on to use the exceptional nature of the alternating group A6 to give a strongly real Beauville structure for this group correcting an earlier error of Fuertes and Gonz´ lez-Diez. In doing so we a complete the classification of alternating groups that possess strongly real Beauville structures. We conclude by discussing mixed Beauville structures of the groups A6 : 2(2) . 1 Introduction 1.1 Beauville Surfaces, Structures and Groups Complex surfaces lie in the intersection of algebraic geometry, differential geometry and complex variable theory and as such enjoy applications as far afield as number theory, topology and even superstring theory. Finding examples of such surfaces that are easy to work with is thus more important than ever. One approach that has proved particularly fruitful over the past ten years or so is the concept of a Beauville surface: a class of 2-dimensional complex algebraic varieties that are rigid, in the sense of admitting no non-trivial deformations, whose study was recently initiated by Bauer, Catanese and Grunewald in [1, 2, 8]. These surfaces are defined over the field Q of algebraic numbers, providing a geometric action of the absolute Galois group Gal(Q/Q). By generalizing Beauville’s original example [4, p.159], they can be constructed from finite groups acting on suitable pairs of algebraic curves, and here we give some new examples of surfaces of this kind. Definition. 1. A Beauville surface of unmixed type is a compact complex surface S such that (a) S is isogenous to a higher product, that is, S (C1 × C2 )/G where C1 and C2 are algebraic curves of genus at least 2 and G is a finite group acting by the diagonal action freely on C1 × C2 by holomorphic transformations; (b) If G0 < G denotes the subgroup consisting of the elements which preserve each of the factors, then G0 acts effectively on each curve Ci so that Ci /G0 P1 and Ci → Ci /G0 ramifies over three points. 1 2 Ben Fairbairn Condition (b) is equivalent to each curve Ci admitting a regular dessin in the sense of the theory of dessin d’enfants due to Grothendieck [9, 19, 36], or equivalently an orientably regular hypermap [23], with G acting as the orientation-preserving automorphism group. One particularly attractive feature of this class of curves is the fact that the above definition can be translated into more finitery combinatorial terms that ‘internalize’ the structure of the surface into the group G in the following way. Definition. 2. Let G be a group. A Beauville structure of G is a pair of generating sets xi , yi , zi ∈ G with li := o(xi ), mi := o(yi ) and ni := o(zi ) for i = 1, 2 such that the following hold. 1. xi yi zi = 1 for each i = 1, 2; 2. l−1 + m−1 + n−1 < 1 for each i = 1, 2 and i i i 3. no non-identity power of x1 , y1 or z1 is conjugate in G to a power of x2 , y2 or z2 . We say that the Beauville structure has type ((l1 , m1 , n1 ), (l2 , m2 , n2 )). We further say that the structure is either mixed or unmixed depending on whether G0 has index 2 or 1 respectively (ie if the corresponding surface is mixed or unmixed). We call a group possessing a Beauville structure a Beauville group. It was conjectured by Bauer, Catanese and Grunewald that every nonabelian finite simple group is a Beauville group, with the sole exception of the alternating group A5 [2, Conjecture 1]. Several authors settled special cases of this conjecture [2, 15, 16, 18]. Finally the full conjecture was recently verified by the author, Magaard and Parker in [13] where we prove the following more general theorem. Theorem 1.1. Aside from the groups SL2 (5) and PSL2 (5)( A5 ) every nonabelian finite quasisimple group possesses an unmixed Beauville structure. 1.2 Real Surfaces and Unmixed Structures Now that we know that almost every quasisimple group is a Beauville group, we are in a position to address the more general issue of what these Beauville structures and surfaces actually look like. One more specific instance of this somewhat vague question is to ask when a ‘complex’ surface S is in fact real (ie there is a biholomorphic map σ : S → S such that σ2 is the identity). As is the ‘zeitgeist’ of Beauville constructions, this topological property can be translated into finitery combinatorial terms inside the corresponding group. Definition. 3. We say that a Beauville surface is real and its corresponding Beauville structure {xi , yi , zi |i = 1, 2} and group G are strongly real if there exist automorphisms φi ∈ Aut(G) for i = 1, 2 φ φ that differ only in an inner automorphism of G such that xi i = x−1 and yi i = y−1 for i = 1, 2. If such i i an automorphism exists we say that G is strongly (li , mi , ni ) generated. It has been conjectured by Bauer, Catanese and Grunewald [1, Conjecture 3] that all but finitely many of the finite simple groups are strongly real Beauville groups. Given the progress made on the wider class of quasisimple groups in theorem 1.1, it seems natural to make following stronger conjecture. Some Exceptional Beauvile Structures 3 Conjecture 1.2. All but finitely many finite quasisimple groups possess strongly real unmixed Beauville structures. Clearly settling the status as strongly real Beauville groups of the sporadic groups makes no impact whatsoever on this conjecture, since there are only finitely many sporadic groups. Nonetheless it is the opinion of the author that settling these questions for the sporadic groups is no less important than, for example, determining their symmetric genii or determining which of them are Hurwitz groups [10, 34]. It is, however, arguably more useful to settle this matter for the sporadic groups since curves and surfaces associated with them are likely to be very exceptional in nature and much of the original motivation for the study of Beauville surfaces was for their use as counterexamples (see, for instance, [1]). To this end we prove the following theorem. Theorem 1.3. (a) The Mathieu groups M11 and M23 are not strongly real Beauville groups. (b) Every other quasisimple sporadic group (except possibly the groups 2˙B and M) is a strongly real Beauville group. Our computationally intensive methods are unable to handle the groups 2˙B and M (though we are able to show that the simple group B is a strongly real Beauville group). We make no apologies for this: to resolve the somewhat analogous problem of settling the Monster’s status as a Hurwitz group took almost ten years of computing time [34, p.370]! Nonetheless, the vast majority of the conjugacy classes in both of these groups are strongly real (see [30]) and since the only problem the groups M11 and M23 encounter is a lack of strongly real classes (see Section 4) we make the following conjecture. Conjecture 1.4. Both of the groups 2˙B and M are strongly real Beauville groups. In the case of M we make several specific conjectural remarks concerning how a strongly real Beauville structure for M might be obtained in Section 5 - the problem is not a lack of theoretical ideas or knowledge about the monster, but simply a lack of computational power! In Section 6 we go on to consider the mixed case where we prove the following theorem. Theorem 1.5. Let G be an almost simple group such that the derived subgroup [G, G] is sporadic. Then G does not possess an unmixed Beauville structure. Finally, our attention turns to the question of which alternating groups possess a strongly real Beauville structure and in doing so we prove the following theorem. Theorem 1.6. The alternating group A6 has a strongly real Beauville structure of type ((4,4,4),(5,5,5)). When combined with the structures explicitly constructed in the proof of [15, theorem 2] we have the following corollary. Corollary 1.7. The alternating group An is a strongly real unmixed Beauville group if and only if n ≥ 6. (Note that in [15, theorem 2] Fuertes and Gonz´ lez-Diez claim to prove the above result with a the bound n ≥ 6 replaced with n ≥ 7 - an error that the above theorem corrects. Interestingly, this 4 Ben Fairbairn correction requires the use of the exceptional nature of Aut(A6 ), so is clearly very different to the n ≥ 7 cases.) We conclude with a brief discussion of mixed Beauville structures of groups of the form A6 : 2(2) . A comprehensive survey article that discusses these and a number of closely related geometric/topological matters is given by Bauer, Catanese and Pignatelli in [3]. 1.3 Our Construction of Strongly Real Beauville Structures Roughly speaking, our method of showing that a group is a strongly real Beauville group, which in principal may be applied to any perfect group of even order that possesses a strongly real Beauville structure, is as follows. Given a group G and an involution t ∈ G we define the elements xi := tt gi for some gi ∈ G for i = 1, 2. Using the observations made as part of Bray’s involution centralizer algorithm [6] we can easily find some involution u ∈ G that commutes with t and does not normalize the subgroup x1 . j(i) We can then define the elements yi := (xi )u for i = 1, 2, the value of the integer j(i) being chosen to make the product xi yi ‘nice’ (ie we ensure that the conditions of definition 2 are satisfied and when necessary hopefully makes it easier to see from the subgroup structure of G that xi , yi = G). This gives a Beauville structure for G. Since u and t commute we also have that xt = x−1 and yt = y−1 i i i i thus the Beauville structure just constructed must be strongly real. Note that we can instead take t to be an element of order 4 that squares to a central involutions. We will not need this alternative approach here. 1.4 Layout of This Paper Throughout, unless otherwise stated, we shall use the standard Atlas notation and conventions for groups and related concepts (accepting the more modern notations B and M for the Baby Monster and the Monster groups respectively) as described in [11]. This paper is organized as follows. In Section 2 we give explicit words in the standard generators using the method of Section 1.3 to define strongly Beauville structures of the groups appearing in part (b) of theorem 1.3 and describe how strongly real Beauville structures for their homomorphic images may be obtained. In Section 3 we prove that the elements given in Section 2 do indeed provide strongly real Beauville groups as claimed. In Section 4 we prove part (a) of theorem 1.3. In Section 5 we discuss the problem of showing that the Monster is a strongly real Beauville group. In Section 6 we prove that no almost simple sporadic group possesses an unmixed Beauville structure and finally in Section 7 we give an explicit strongly real Beauville structure for the alternating group A6 proving theorem 1.6. Acknowledgments The author wishes to express his deepest gratitudes to Professor Gareth Jones for first introducing him to the concept of ‘all things Beauville’; to Dr John Bradley and Professor Christopher Parker for providing many helpful comments and corrections to early versions of this paper and to Dr Matthew Fayers for assistance with the typesetting. Some Exceptional Beauvile Structures G J1 2˙M12 12˙M22 2˙J2 2˙HS 3˙J3 M24 3˙McL He 2˙Ru 6˙Suz 3˙O’N Type ((19,19,11),(15,15,7)) ((5,5,3),(11,11,11)) ((5,5,5),(12,12,6)) ((7,7,7),(12,12,8)) ((15,15,5),(8,8,7)) ((17,17,19),(9,9,9)) ((5,5,5),(6,6,11)) ((5,5,5),(6,6,6)) ((3,3,6),(17,17,17)) ((4,4,10),(13,13,7)) ((13,13,13),(12,12,10)) ((28,28,12),(19,19,19)) G Co3 Co2 6˙Fi22 HN Ly Th Fi23 2˙Co1 J4 3˙Fi24 B Type ((7,7,23),(5,5,24)) ((11,11,14),(7,7,15)) ((7,7,5),(13,13,13)) ((5,5,5),(6,6,6)) ((67,67,40),(37,37,21)) ((19,19,19),(13,13,13)) ((5,5,5),(6,6,4)) ((5,5,5),(6,6,6)) ((43,43,11),(29,29,6)) ((9,9,9),(11,11,26)) ((13,13,19),(12,12,20)) 5 Table 1: The types of the Beauville structures defined by the words given in Tables 2 and 3. See Definition 1.1. 2 Strongly Real Sporadic Beauville groups 2.1 The Structures In this section we describe and tabulate the Beauville structures that we construct here. Standard generators for the sporadic groups are given on the online atlas [35] and are named a and b. In each case we have that o(a) = 2, so where possible we use this element to define the automorphisms needed when constructing strongly real Beauville structures. For background information on standard generators more generally see the original article by Wilson [31]. The types of the Beauville structures we construct here are given in Table 1. The words used to define our Beauville structures are given in Table 2. We remark that whilst it is common to use lower case letters for the standard generators of a simple group and upper case letters for their covering groups. For the sake of aesthetics we use lower case letters in both cases, it being clear which are the non-simple cases. In some cases it is either necessary or desirable to use an involution other than a that we call c. The words in the standard generators used to define these elements c are given in Table 3. In each case the fact that the given elements generate may be verified using either permutation or matrix representations of these groups available on [35], either directly or by the observations made in the next section. 6 Ben Fairbairn G J1 2˙M12 12˙M22 2˙J2 2˙HS 3˙J3 M24 3˙McL He 2˙Ru 6˙Suz 3˙O’N Co3 Co2 6˙Fi22 HN Ly Th Fi23 2˙Co1 J4 3˙Fi24 B x1 aab aa(ba) b ab 2 aabab 2 2 4 aa(b a) b 3 ccb a 3 aa(ba) b aabab 2 aa(ba) b 3 ccb ccaba 7 cc(ba) aabab 2 aa(bab)b 2 ccb 7 ccbab aab 2 aa(ba) b 2 b2 aa(ba) 2 aab ab 2 2 cc(ab ab) a 2 3 ccab ab ab ccb 2 2 aa(ba) b 2 2 2 x2 aabab 2 aa(bab) ab aab 2 4 aa(ba) b 3 ccab ab aab 2 aabab 2 4 aa(ba) b ccbab 3 ccabab a ccb 2 2 aab ab 2 2 aab abab ab 2 2 ccb ab ab 2 4 ccb ab aababab 2 2 aa(ba) b ab 4 2 aa(ba) b ab 3 2 aa(ba) b 2 2 ccab (ab) a 3 4 cc(ab ) 3 cc(ab) 2 b2 ab aa(ba) u b(ab2 ab)9 bab(a(bab)23 (ab)2 )2 b2 ab(ab(b2 a)3 b)2 (a(ab2 )23 a2 b2 )6 (cb3 cb2 )6 b(ab2 ab)4 (ab2 ab)6 b(ab4 ab)2 (cb6 cb)2 a(ca3 ca)3 (ca)20 (ab3 ab)6 b(ab3 ab)7 (cb4 cb)15 (cb12 cb)10 b(ab2 ab)2 b(ab4 ab)4 (ab2 ab)5 b(ab2 ab)2 b(cb2 cb)2 b(cb3 cb)15 b(cb2 cb)4 (ab2 ab)20 j(1) 8 4 3 1 14 3 1 1 1 3 6 2 5 1 6 1 3 1 2 2 1 1 2 j(2) 10 3 5 3 1 1 1 1 13 2 1 12 1 3 6 1 2 3 1 1 1 5 1 Table 2: Words in terms of the standard generators defining a strongly real Beauville structure for the full covering group of each of the sporadic simple groups considered here. In each case the elements a and b are the standard generators. In cases where the use of an element labeled c is required, words in the standard generators defining these elements are given in Table 3. In some cases it was necessary/desirable to use the standard generators for G : 2 rather than G. These cases we write in bold font. Some Exceptional Beauvile Structures G 2˙HS He 2˙Ru 6˙Suz Co2 c (bab2 ab4 a)5 (ab3 )4 b2 (bab2 (ab)2 )28 ab(ba)2 b((bab)2 (ab2 )2 )2 G 6˙Fi22 2˙Co1 J4 3˙Fi24 c (abab10 )6 (ab)20 (abab2 )5 ((ab)4 b)18 7 Table 3: Words in terms of the standard generators a and b defining an involution c in cases the involution a is unable to define a strongly real Beauville structure via the construction described in Section 1.3. 2.2 Homomorphic images So far we have only shown that the full covering group of each of the groups of part (b) of Theorem 1.3 are strongly real Beauville groups. In this subsection we consider the cases of the quasisimple sporadic groups with non-trivial centers and their homomorphic images. Given a group G, it is tempting to look for a Beauville structure in the quotient G/N by some normal subgroup N G, and to try to lift this back to G. However, a triple that generates G/N need not lift back to a triple generating G, and even if it does, the condition (2) of definition 2 may not be satisfied. In this situation, the following two lemmata are of great use (whilst the proofs of these results may seem trivial to the group theorist we include references to their proofs for the sake of the less group theoretically inclined reader). Lemma 2.1. If G is a perfect group, N is a central subgroup of G, and S is a subset of G such that the image of S in G/N generates G/N, then S generates G. Proof. See [16, Lemma 4.1]. Lemma 2.2. Let G have generating triples (xi , yi , zi ) with xi yi zi = 1 for i = 1, 2, and a normal subgroup N such that at least one of these triples is faithfully represented in G/N. If the images of these triples correspond to a Beauville structure for G/N, then these triples correspond to a Beauville structure for G. Proof. See [16, Lemma 4.2]. From the types of the Beauville structures obtained in the previous section and from the orders of the centers of the relevant groups it is clear that the above lemmata may be applied to the Beauville structures obtained in the previous section. 3 Large Strongly Real Beauville Groups In this Section we prove that the Beauville structures defined in Section 2 do indeed generate the groups claimed in the cases where the representations of the groups in question are too cumbersome for this to be verified directly. In doing so we complete the proof of part (b) of theorem 1.3. In each case it is taken for granted that the elements refered to from the previous section do indeed have the stated orders and we focus only on the question of generation in each case. Any direct calculation refered to in the below proofs may easily be performed in Magma [5] or GAP [17]. 8 Ben Fairbairn Lemma 3.1. The Harada-Norton group HN possosses a strongly real Beauville structure of type ((5,5,5),(6,6,6)). Proof. From the list of maximal subgroups of HN, as listed in the Atlas [11, p.166], we see that no proper subgroup contains elements of order 22 and order 25. Direct computation shows that o(x1 y1 x2 y3 ) = o(x2 y3 x2 y4 ) = 22 and o(x1 y1 x3 y4 ) = o(x2 y2 x4 x5 ) = 25, hence x1 , y1 = x2 , y2 = G. 2 2 2 2 2 1 1 1 1 Lemma 3.2. The Lyons group Ly possesses a strongly real Beauville structure of type ((67,67,40),(37,37,21)). Proof. From the list of maximal subgroups of Ly, as listed in the Atlas [11, p.174], we see that an element of order 67 is contained in only one maximal subgroup, a copy of the Frobenious group 67:22. This clearly contains no elements of order 40. Similarly we see that an element of order 37 is contained in only one maximal subgroup, a copy of the Frobenious group 37:18. Since this clearly contains no elements of order 21 we must have x1 , y1 = x2 , y2 = G. Lemma 3.3. The Thompson group Th possesses a strongly real Beauville structure of type ((19,19,19),(13,13,13)). Proof. From the list of maximal subgroups of Th, as listed in [24, 25] (note that list given in the Atlas [11, p.177] is incomplete), we see that the only maximal subgroups containing elements of order 31 are isomorphic to either 25 ˙L5 (2) or the Frobenious group 31:15. These subgroups clearly contain no elements of order 19 or 13. Direct computation shows that o(x1 y1 x2 y4 ) = o(x2 y2 x2 y11 ) = 31, and 2 2 1 1 so x1 , y1 = x2 , y2 = G. Lemma 3.4. The Janko group J4 possesses a strongly real Beauville structure of type ((43,43,11),(29,29,6)). Proof. From the list of maximal subgroups of J4 , as listed in the Atlas [11, p.190], we see that an element of order 43 is contained in only one maximal subgroup, a copy of the Frobenious group 43:14. This clearly contains no elements of order 11. Similarly we see that an element of order 29 is contained in only one maximal subgroup, a copy of the Frobenious group 29:28. Since this clearly contains no elements of order 6 we must have x1 , y1 = x2 , y2 = G. Lemma 3.5. The Baby Monster B possesses a strongly real Beauville structure of type ((13,13,19),(12,12,20)) Proof. From the list of maximal subgroups of B, as listed in [32] (note that list given in the Atlas [11, p.217] is incomplete), we see that the only maximal subgroup containing elements of order 47 is isomorphic the Frobenious group 47:23. This subgroup clearly contains no elements of order 13 or 12. Direct computation shows that o(x1 y7 x3 y5 ) = o(x2 y2 x3 y7 ) = 47, and so x1 , y1 = x2 , y2 = G. 2 2 1 1 1 4 Non-Strongly Real Beauville Groups In this short section we prove that the sporadic groups M11 and M23 ) are not strongly real Beauville groups. In doing so we complete the proof of theorem 1.3. Note that the strongly real classes of the sporadic simple group were classified by Suleiman in [30]. Lemma 4.1. The groups M11 and M23 are not strongly real Beauville groups. Some Exceptional Beauvile Structures Element p qr qsr Order 42 19 35 Element s q2 r 2 qs2 r Order 39 57 105 9 Table 4: Some elements of M inverted by conjugation by g and their orders. (x,y,xy) (p, s, ps) (p2 , s, p2 s) (p3 , s, p3 s) (p, qr, pqr) (p, qsr, pqsr) (o(x), o(y), o(xy)) (42,39,19) (21,39,39) (14,39,56) (42,19,42) (42,35,57) (x,y,xy) (qr, s, qrs) (qr, s2 , qrs2 ) (p2 , qr, p2 qr) (p2 , s2 , p2 s2 ) (qsr, s, qsrs) (o(x), o(y), o(xy)) (19,39,22) (19,39,66) (21,19,60) (21,39,55) (35,39,105) Table 5: Some sets of elements of M that could potentially strongly (a, b, c)-generate the group. Proof. In both cases the only strongly real classes are classes of elements of order at most 6. In both groups there is only one class of elements of orders 2, 3 or 5. Computer calculations show that in each case, the group is only strongly (5, 5, m) generated if the integer m is 4 or 6 and that neither group is strongly (3, 3, m) generated for any integer m. We remark that in [1, p.35] Bauer, Catanese and Grunewald state that they were unable to find a strongly real Beauville structure for M11 (among other groups). The above lemma explains why. 5 The Monster We give a brief discussion as to how a strongly real Beauville structure of the monster group M might be obtained. In [34] Wilson proves that M can be generated by a pair of elements g and h such that g is in class 2B, h is in class 3B and gh is in class 7B. In the process of proving this Wilson defines the following four elements p = ghgh2 , q = ghghgh2 , r = ghgh2 gh2 , s = ghghgh2 gh2 . Firstly, to apply our construction of Section 1.3 we need an involution of M - naturally we take the element g. The orders of several short words in the elements above are given in [34, Table 1]. In particular we have that o(p) = 42. Now, for our Bray-type element, u, observe that p g = p−1 and so g p21 ∈ Z( p, g ). Whilst other short words in the above elements, such as those appearing in Table 4, are inverted by conjugation by g, these words often have odd order and so there is no guarantee that the involution produced will be distinct from g. For our elements xi , i = 1, 2 (which immediately give us the elements yi := xu ) we note that i several of the words given in [34, Table 1] are inverted by conjugation by g, such as those given in Table 4 and their powers, and any one of these provide candidates for our xi s. 10 Ben Fairbairn A slightly different approach is as following. In several cases, the products of elements found in Table 4 also have their orders listed in [34, Table 1]. We can thus define at least one of our (potential) generating pairs by taking these elements themselves. We list a few of these possibilities in Table 5. We remark that proving that a proposed generating set M does in fact generate is easier than it first appears. Whilst the maximal subgroups of M have yet to be classified, a substantial amount of information is known. In particular, a complete classification of the maximal subgroups that contain elements of class 2A is known - see [27]. An immediate corollary of this classification is that the only maximal subgroups of M containing elements of order 94 are copies of 2˙B. Finding a word in our set of proposed generators of order 94 forces the set to be contained in some copies of 2˙B and another word in our set of proposed generators that cannot lie in such a subgroup proves that the set generates. This is precisely how Wilson showed in [34] that the above g and h generate M - it turns out that o(ppqsrpsrqsq) = 94 and o(ppqsrqqrprq) = 41. Whilst multiplying elements of M together is extremely difficult, computing the order of such a word is somewhat easier - the method described in [26], computing orders by analyzing orbits of specially chosen vectors in the natural 196882 dimensional F2 module, being the method used to calculate the orders given above. 6 Mixed Beauville structures In this short section we consider the unmixed case and prove theorem 1.5. Recall that a Beauville structure is said to be mixed if the setwise stabalizer of the curves defining the corresponding surface, G0 ≤ G, has index 2. Clearly no simple group can possess an unmixed Beauville structure, however this doesn’t rule out the possibility of an almost simple group possessing one. The following easy lemma is extremely useful. Lemma 6.1. Let (C × C)/G be a Beauville surface of mixed type and G0 the subgroup of G consisting of the elements which preserve each of the factors, then the order of any element in G \ G0 is divisible by 4. Proof. See [15, Lemma 5]. Of the 26 sporadic groups twelve of them (namely M12 , M22 , J2 , HS, J3 , McL, He, Suz, O’N, Fi22 , HN and Fi24 ) posses outer automorphisms. From their character tables, which can be reconstructed from the data given in [11], we see that all of the almost simple groups whose derived subgroup is in the above list have involutions lying outside G0 and so by the above lemma none of these groups can possess a mixed Beauville structure. This proves theorem 1.5. 7 The Alternating Groups In this final section we prove theorem 1.6 and corollary 1.7. To do this we first recall some standard facts about automorphisms of alternating groups. If n 2, 3 or 6 then Aut(An ) Sn , the symmetric group. (If n = 2, 3 then Aut(An ) Sn−1 .) If n = 6 then we have that S6 is an index 2 subgroup of Aut(A6 ) and this was first proved as long ago as Some Exceptional Beauvile Structures 11 1895 by Otto Holder in [20]. There are many different ways to prove this and almost any good book ¨ on finite group theory as well as numerous other sources will present at least one approach (for example [7, Chapter 6], [12], [14], [21], [22], [28, Problem 4.20] [29, p.156-162], [33, Sections 2.4.2, 3.3.5 & 4.2]). An immediate consequence of this fact is the result that Aut(A6 ) has three index 2 subgroups, each of structure A6 : 2. One is isomorphic to the linear group PGL2 (9) (the exceptional isomorphism A6 PSL2 (9) gives us the fact that Aut(A6 ) PΓL2 (9)); another to S6 PΣL2 (9) and the final one to the Mathieu group M10 . Proof. of theorem 1.6. Consider the permutations x1 := (2, 9, 5, 6)(3, 4, 7, 8), y1 := (1, 3, 8, 5)(2, 6, 10, 4), x2 := (1, 9, 4, 6, 2)(3, 5, 7, 10, 8), y2 := (1, 3, 2, 5, 7)(4, 8, 6, 10, 9), and g := (1, 10)(2, 8)(3, 6)(4, 5)(7, 9). Easy calculation gives o(x1 ) = o(y1 ) = o(x1 y1 ) = 4 and o(x2 ) = o(y2 ) = o(x2 y2 ) = 5. Easy computations further show that x1 , y1 = x2 , y2 = A6 . From their orders it is clear that these elements also satisfy conditions 2 and 3 of definition 1 and so these permutations define a Beauville structure for A6 of type ((4,4,4),(5,5,5)). We claim that this Beauville structure is strongly real. Easy computations show that x1 , y1 , g = x2 , y2 , g = PGL2 (9), one of the groups of the form A6 : 2 not isomorphic to the symmetric group S6 (or the Mathieu group M10 ). Further direct g g calculation reveals that xi = x−1 and yi = y−1 for i = 1, 2 and so this (outer) automorphism of A6 i i shows that this Beauville structure is strongly real. Proof. of corollary 1.7. For n ≥ 7 these are explicitly constructed in the proof of [15, theorem 2]. If n = 6 this is the above theorem. If n ≤ 5 then it is easily verified that An does not even possess a Beauville structure let alone a strongly real one. In [15] Fuertes and Gonz´ lez-Diez use lemma 6.1 to show that S6 does not possess a mixed a Beauville structure. In the case of PGL2 (9) there are involutions lying outside the derived subgroup and so this same lemma ensures that PGL2 (9) also does not possess a mixed Beauville structure. Remarkably, in the case of the group M10 the only elements lying outside the derived subgroup all have order 4 or 8, so lemma 6.1 is of no use here. There is, however, only one class of involutions in M10 and so the group cannot possess any Beauville structure, mixed or not, since condition 3 of definition 2 cannot be satisfied. The group PΓL2 (9) also cannot have an mixed Beauville structure since for each of the index 2 subgroups there is a class of involutions lying outside the subgroup blocking the existence of a mixed Beauville structure by lemma 6.1. It follows that no group of the form A6 : 2(2) possesses a mixed Beaville structure. References [1] I.C. Bauer, F. Catanese and F. Grunewald “Beauville surfaces without real structures I” in Geometric Methods in Algebra and Number Theory, Progr. Math. 235, Birkh¨ user Boston, Boston, a 2005, pp. 1-42 12 Ben Fairbairn [2] I.C. Bauer, F. Catanese and F. Grunewald “Chebycheff and Belyi polynomials, dessins denfants, Beauville surfaces and group theory”, Mediterr. J. Math. 3 (2006), 121-146 [3] I.C. Baur, F. Catanese and R. Pignatelli “Complex surfaces of general type: some recent progress”, in Global Aspects of Complex Geometry, (eds. F.Catanese et al.) 1–58 Springer (Berlin, Heidelberg) (2006) [4] A. Beauville “Surfaces alg´ briques complexes”, Ast´risque 54, Soc. Math. France, Paris, 1978 e e [5] W. Bosima, J. Cannon and C. Playoust “The Magma algebra system I. The user language”, J. Symbolic Comput. 24 (3-4): 235–265, 1997 [6] J.N. Bray “An improved method for generating the centralizer of an involution”, Arch. Math. 74 (2000) 241–245 [7] P.J. Cameron and J.H. van Lint “Graphs, Codes and Designs” LMS lecutre note series 43, CUP (1980) [8] F. Catanese “Fibred surfaces, varieties isogenous to a product and related moduli spaces”, Am. J. Math. 122 (2000), 1-44 [9] P.B. Cohen, C. Itzykson and J. Wolfart “Fuchsian triangle groups and Grothendieck dessins: Variations on a theme of Bely˘ Comm. Math. Phys. 163 (1994), 605627 i”, [10] M.D.E. Conder, RA Wilson and AJ Woldar “The symmetric genus of sporadic”, Proc. Amer. Math. Soc., 116 (1992), no. 3, 653–663 [11] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker and R.A. Wilson “An ATLAS of finite groups” (Oxford University Press, 1985) [12] R.T. Curtis “A fresh approach to the exceptional automorphism and covers of the symmetric groups”, The Arabian Journal for Science and Engineering, Volume 27, Number 1A (2002) [13] B.T. Fairbairn, K. Magaard and C.W. Parker “Quasisimple groups, triangle groups and an application to Beauville structures”, preprint 2010 [14] T.A. Fornelle “Symmetries of the Cube and Outer automorphisms of S6 ”, American Mathematical Monthly, 100(4), (1993), 377–380 [15] Y. Fuertes and G. Gonz´ lez-Diez “On Beauville structures on the groups Sn and An ”, Math. Z. a 264, No. 4, 959-968 (2010) [16] Y. Fuertes and G. Jones “Beauville surfaces and finite groups”, preprint 2009, available at the time of writing from http://arxiv.org/abs/0910.5489 [17] The GAP Group, GAP — Groups, Algorithms, and Programming, Version 4.4.12; 2008 (http://www.gap-system.org) [18] S. Garion and M. Penegini “New Beauville surfaces, moduli spaces and finite groups”, preprint 2009, available at the time of writing from http://arxiv.org/abs/0910.5402 Some Exceptional Beauvile Structures 13 [19] A. Grothendieck “Esquisse dun Programme”, pp. 5-84 in Geometric Galois Actions 1. Around Grothendiecks Esquisse dun Programme, ed. P. Lochak, L. Schneps, London Math. Soc. Lecture Note Ser. 242, Cambridge University Press, 1997 [20] O. Holder “Bildung zusammengesetzter Gruppen”, Math. Ann., 46 (1895) 321–422 ¨ [21] B. Howard, J. Millson, A. Snowden and R. Vakil “A description of the outer automorphism of S6 , and the invariants of six points in projective space”, J. Combin. Theory Ser. A 115 (2008), no. 7, 1296–1303 [22] G. Janusz and J. Rotman “Outer automorphisms of S6 ”, American Mathematical Monthly, 89(6),(1982), 407-410 [23] G.A. Jones and D. Singerman “Belyi functions, hypermaps and Galois groups”, Bull. London Math. Soc. 28 (1996), 561-590 [24] S.A. Linton “The maximal subgroups of the Thompson group”, J. London Math. Soc. (2) 39 (1989), no. 1, 79–88 [25] S.A. Linton “Corrections to: The maximal subgroups of the Thompson group”, J. London Math. Soc. (2) 43 (1991), no. 2, 253–254 [26] S.A. Linton, R.A. Parker, P.G. Walsh and R.A. Wilson “Computer construction of the Monster”, J. Group Theory 1 (1998), 307–337 [27] S.P. Norton and R.A. Wilson “Anatomy of the Monster: II”, Proc. London Math. Soc. (3) 84 (2002) 581–598 [28] H.E. Rose “A Course on finite groups” Springer (2010) [29] J.J. Rotman “An introduction to the theory of groups” Springer (1994) [30] I. Suleiman “Strongly real elements in sporadic groups and alternating group” Jordan Journal of Mathematics and Statistics 1(2) (2008), 97–103 [31] R.A. Wilson “Standard generators for the sporadic simple groups” J. Algebra 184, (1996) 505–515 [32] R.A. Wilson “The Maximal Subgroups of the Baby Monster, I”, J. Algebra 211, (1999) 1–14 [33] R.A. Wilson “The Finite Simple Groups”, GTM 251, Springer, (2009) [34] R.A. Wilson “The Monster is a Hurwitz group” J. Group Theory 4 (2001) 367–374 [35] R.A. Wilson et al “Atlas of Finite Group Representations - Version 3”, available at the time of writing from http://brauer.maths.qmul.ac.uk/Atlas/v3 [36] J. Wolfart “ABC for polynomials, dessin denfants, and uniformization a survey”, pp. 313-345 in Elementare und Analytische Zahlentheorie (Tagungsband), Proceedings ELAZConference May 2428, 2004 (ed. W. Schwarz, J. Steuding), Steiner Verlag Stuttgart 2006 (available at the time ofwriting from http://www.math.uni-frankfurt.de/∼wolfart/).