We discuss Beauville groups whose corresponding Beauville surfaces

are either always strongly real or never strongly real producing several

infinite families of examples

A Beauville

BT Fairbairn

D Singerman

E Pierro

F Catanese

G González-Diez

GA Fernández-Alcober

GA Jones

GA Miller

GV Belyĭ

JH Conway

M Aschbacher

M Suzuki

MDE Conder

NW Barker

R Steinberg

S Garion

T Dedieu

Y Fuertes

arXiv

We discuss Beauville groups whose corresponding Beauville surfaces are either always strongly real or never strongly real producing several infinite families of examples

Fairbairn, Ben

Name not available

Purely (Non-)Strongly Real Beauville Groups

Beauville surfaces are a class of complex surfaces defined by letting a finite group G act on a product of Riemann surfaces. These surfaces possess many

attractive geometric properties several of which are dictated by properties of the group G. A particularly interesting subclass are the ‘strongly real’ Beauville surfaces that have an analogue of complex conjugation defined on them. In this survey we discuss these objects and in particular the groups that may be used to define them. En route we discuss several open problems, questions and conjectures and in places make some progress made on addressing these

Birkbeck Institutional Research Online

Strongly Real Beauville Groups III

BIROn - Birkbeck Institutional Research OnlineFairbairn, Ben (2019) Purely (Non-)Strongly Real Beauville Groups. Archivder Mathematik 112 (4), pp. 337-345. ISSN 0003-889X.Downloaded from: http://eprints.bbk.ac.uk/24840/Usage Guidelines:Please refer to usage guidelines at http://eprints.bbk.ac.uk/policies.html or alternativelycontact lib-eprints@bbk.ac.uk.Purely (Non-)Strongly Real Beauville GroupsBen FairbairnAbstract. We discuss Beauville groups whose corresponding Beauvillesurfaces are either always strongly real or never strongly real producingseveral infinite families of examples.Mathematics Subject Classification (2010). Primary 20D06; Secondary14J29, 30F10.Keywords. Beauville surface, Strongly real surface, Beauville group.1. IntroductionDefinition 1. A surface S is a Beauville surface if• the surface S is isogenous to a higher product, that is, S ∼= (C1 ×C2)/Gwhere C1 and C2 are algebraic curves of genus at least 2 and G is a finitegroup acting faithfully on C1 and C2 by holomorphic transformations insuch a way that it acts freely on the product C1 × C2, and• each Ci/G is isomorphic to the projective line P1(C) and the coveringmap Ci → Ci/G is ramified over three points.These surfaces were first defined by Catanese in [3] and the first signifi-cant investigation of them was conducted by Bauer, Catanese and Grunewaldin [2]. They have numerous nice properties and are relatively easy to constructmaking them useful for producing counterexamples and testing conjectures.The following condition is also investigated in [2].Definition 2. Let S be a complex surface. We say that S is strongly real ifthere exists a biholomorphism σ : S → S such that σ ◦ σ is the identity map.What makes these surfaces particularly easy to work with is that all ofthe above can be easily translated into group theoretic terms.Definition 3. Let G be a finite group. Let x, y ∈ G and letΣ(x, y) :=|G|⋃i=1⋃g∈G{(xi)g, (yi)g, ((xy)i)g}.2 Ben FairbairnA Beauville structure for the group G is a set of pairs of elements{(x1, y1), (x2, y2)} ⊂ G × G with the property that 〈x1, y1〉 = 〈x2, y2〉 = Gsuch thatΣ(x1, y1) ∩ Σ(x2, y2) = {e}.If G has a Beauville structure we say that G is a Beauville group.A group defines a Beauville surface if and only if it has a Beauville struc-ture. Furthermore, the Beauville surface defined by a particular Beauvillestructure is strongly real if and only if the corresponding Beauville structurehas the property of being strongly real that we define as follows.Definition 4. Let G be a Beauville group and let X = {(x1, y1), (x2, y2)} bea Beauville structure for G. We say that G and X are strongly real if thereexists an automorphism φ ∈ Aut(G) and elements gi ∈ G for i = 1, 2 suchthatgiφ(xi)g−1i = x−1i and giφ(yi)g−1i = y−1i .Most of the Beauville structures appearing in the literature are eitherexplicitly shown to be strongly real or the question of reality is never pursued.In many groups there are elements that can never be inverted by automor-phisms meaning any Beauville structure defined using such elements cannotbe strongly real, indeed groups typically have numerous Beauville structuressome of which are strongly real, some of which are not. Here we are interestedin the extreme cases and thus make the following definition.Definition 5. A finite group G is a purely strongly real Beauville group ifG is a Beauville group such that every Beauville structure of G is stronglyreal. A finite group G is a purely non-strongly real Beauville group if G isa Beauville group such that none of its Beauville structures are strongly real.Throughout we shall follow the conventions that in a group G and ele-ments g, h ∈ G we have that gh = hgh−1 and [g, h] = ghg−1h−1.This paper is organised as follows. In Section 2 we will give infinitelymany examples of Beauville groups that have strongly real Beauville struc-tures as well as Beauville structures that are not suggesting that groups typ-ically lie in neither of the categories in Definition 5. Despite this, we go on inSection 3 to give infinitely many examples of purely strongly real Beauvillegroups before in the final section giving infinitely many examples of purelynon-strongly real Beauville groups.2. Neither CaseWe first prove results showing that Beauville groups typically fall into neithercase with several different examples.Lemma 6. If n > 5, then the alternating group An is neither a Purely stronglyreal Beauville group nor a non-strongly real Beauville group.Purely (Non-)Strongly Real Beauville Groups 3Proof. Strongly real Beauville structures for these groups were constructedby Fuertes and Gonza´lez-Diez in [8] so it is sufficient to construct non-stronglyreal Beauville structures for these groups. We consider the cases n odd and neven separately. Recall that in either case, if two permutations have differentcycle type, then they cannot be conjugate.First suppose that n ≥ 7 is odd. Letx1 := (1, 2, 4) and y1 := (1, 2, 3, 4, . . . , n).We have that their product is the n-cycle x1y1 = (1, 3, 4, 2, 5, . . . , n) by directcalculation. By considering the subgroup generated by the elements xyi11 wehave that the subgroup these elements generate is 2-transitive and thereforeprimitive. Since this subgroup contains the 3-cycle x1 it follows that theseelements generate the whole of the alternating group. It is easy to see thatno automorphism simultaneously inverts both of these elements so any per-mutations that can be used to extend this to a Beauville structure gives anon-strongly real Beauville structure. Consider the permutationsx2 := (5, 4, 3, 2, 1) and y2 := (3, 4, 5, 6, . . . , n).We have that their product is the (n − 2)-cycle x2y2 = (1, 6, . . . , n, 3, 2) bydirect calculation. By considering the subgroup generated by the elementsxyi22 we have that the subgroup these elements generate is 2-transitive andtherefore primitive. Since this subgroup contains the double-transposition[x2, y2] = (1, 5)(3, n) it follows that these elements generate the whole of thealternating group. We therefore have a non-strongly real Beauville group.Next, suppose n ≥ 8 is even. Letx1 := (1, 2)(3, 4) and y1 := (2, 3, 4, . . . , n).We have that their product is the (n − 1)-cycle x1y1 = (1, 3, 5, . . . , n, 2) bydirect calculation. By considering the subgroup generated by the elementsxyi11 we have that the subgroup these elements generate is 2-transitive andtherefore primitive. Since this subgroup contains the double-transposition x1it follows that these elements generate the whole of the alternating group.It is easy to see that no automorphism simultaneously inverts both of theseelements so any permutations that can be used to extend this to a Beauvillestructure gives a non-strongly real Beauville structure.Finally, letx2 := (1, 2, 3)(4, 5, . . . , n) and y2 := (5, 4, 3, 2, 1).We have that their product is the (n−3)-cycle x2y2 = (3, 5, . . . , n). By consid-ering the subgroup generated by the elements yxi22 we have that the subgroupthese elements generate is 2-transitive and therefore primitive. Since this sub-group contains the double transposition [x2, y22 ] = (2, 5)(3, n) it follows thatthese elements generate the whole of the alternating group. We therefore havea non-strongly real Beauville group.The case of A6 is complicated by the existence of exceptional outerautomorphisms but despite this can easily be handled separately. 4 Ben FairbairnFurther examples are given by the following.Lemma 7. None of the Suzuki groups 2B2(22n+1) are purely (non-)stronglyreal Beauville groups.Proof. The Beauville structures constructed by Fuertes and Jones in [9, The-orem 6.2] are non-strongly real since they take y1 as having order 4 and noautomorphism maps such elements to their inverses in these groups. Stronglyreal Beauville structures for these groups were constructed by the author in[5]. Lemma 8. Aside from the Mathieu groups M11 and M23 none of the sporadicsimple groups are purely (non-)strongly real Beauville groups.Proof. Aside from M11 and M23, strongly real Beauville structures for thesegroups were constructed by the author in [4]. Non-strongly real Beauvillestructures are easily obtained computationally and with character theory,the more difficult cases being easily dealt with thanks to the existence ofelements that cannot be sent to their inverses by any automorphism at allsuch as elements of order 71 in the monster group M and elements of order47 in the baby monster group B. We will return to the cases of M11 and M23 in Section 4 since they aregenuine exceptions to the above.It is clear that numerous further examples can be constructed from theabove using direct products.3. Purely Strongly Real Beauville GroupsFirstly, the following observation has been made elsewhere in the litera-ture many times and provides infinitely many purely strongly real Beauvillegroups.Lemma 9. A finite abelian group is a strongly real Beauville group if and onlyif it is a purely strongly real Beaville group.Proof. In any abelian group the homomorphism x 7→ −x inverts every ele-ment. Abelian Beauville groups were first constructed by Catanese in [3] andclassified by Bauer, Catanese and Grunewald in [2]. For non-soluble examples,we have the following infinite supply.Proposition 10. Let q > 2 be a power of 2 and let k be a positive integer.If (q, k) 6= (4, 1), then the characteristically simple group L2(q)k is a purelystrongly real Beaville group whenever it is 2-generated.Proof. In [13] MacBeath observed that any generating pair for the groupsL2(q) can be inverted by an inner automorphism when q is even and sincethese groups have only one class of involutions, the only elements of evenorder. Purely (Non-)Strongly Real Beauville Groups 5We remark that the exception (q, k) = (4, 1) is a genuine exceptionthanks to the isomorphism A5 ∼= L2(4) and this group is well known to notbe a Beauville group.For each prime p ≥ 5 we can also construct infinitely many (new) non-abelian nilpotent examples as follows.Proposition 11. If p ≥ 5 is a prime and n ≥ r ≥ 1 are integers, then thegroupG := 〈x, y, z |xpn , ypn , zpr [x, y] = z, [x, z], [y, z]〉is a purely strongly real Beauville group.Proof. Let {{x1, y1}, {x2, y2}} be a Beauville structure of G. We first notethat Aut(G) acts transitively on the non-central elements ofG of a given orderand a generating pairs must consist of two element of order pn, so withoutloss we may assume that x1 = x. Similarly CAut(G)(x) acts transitively theelements x can generate with so without loss we may also assume that y1 = yand that φ ∈Aut(G) acts by xφ = x−1 and yφ = y−1 (so we can take theelement g1 of Definition 4 to be trivial).Observe the following. For any element g ∈ G \ Φ(G) we have that itsconjugates are gG = {gzi | i = 0, . . . , pr−1}. Every element of this group canbe written in the form xiyjzk for some 0 ≤ i, j ≤ pn − 1 and 0 ≤ k ≤ pr − 1,so we have that x2 = xi1yj1zk1 and y2 = xi2yj2zk2 for some 1 ≤ i1, i2, j1, j2 ≤pn−1 and 0 ≤ k1, k2 ≤ pr−1. Notice that if these are chosen so that i1 6= j1,i2 6= j2, i1 + i2 6= j1 + j2 and o(x2) = o(y2) = pn, then these elements providea Beauville structure and since p ≥ 5 finding such integers is straightforward.Moreover(xiyjzk)φ = x−iy−jzk and (xiyjzk)−1 = x−iy−jz−k−ijWe also have that for any 0 ≤ a, b ≤ pn − 1(x−iy−jzk)xayb = x−iy−jzk+bi−aj .It follows that to have (xφ2 )xayb = x−12 and (yφ2 )xayb = y−12 we must have thatbi1 − aj1 ≡ −2k1 − ij (mod pr) and bi2 − aj2 ≡ −2k2 − ij (mod pr).If the values of i1, i2, j1, j2, k1 and k2 are chosen so x2 and y2 generate thegroup and the conjugacy condition is satisfied, then values of a and b satis-fying these equations can be found if j2i1 6≡ j1i2 (mod p). We claim that ifj2i1 ≡ j1i2 (mod p), then x2 and y2 do not generate the group. Note thatunder this condition we have that(xi1yj1)i2 = xi1i2yj1i2zk = xi1i2yj2i1zkfor some k but we also have that(xi2yj2)i1 = xi1i2yj2i1zk′for some k′ in other words y2 ∈ 〈x2, z〉 which is a proper subgroup. 6 Ben FairbairnWe remark that strongly real Beauville p-groups have proved somewhatdifficult to construct the only previously known examples being given in [6,7, 10, 11]. The above provides infinitely many further new examples and doesso for each prime p ≥ 5.Clearly Proposition 10, Lemma 9 and Proposition 11 can be combinedto produce infinitely many other examples like L2(8)×C25 but it is not clearwhat other examples can arise.Problem 12. Find other examples of purely strongly real Beauville groups.In particular, we have the following question.Question 13. Do there exist purely strongly real Beauville 2-groups and 3-groups.In the opinion of the author it seems likely that 2-generated 2-groupsare more likely to be purely strongly real Beauville groups: there is a generalphilosophy in the study of p-groups that ‘the automorphism group of a p-group is typically a p-group’ thanks to the results of Helleloid and Martin[12]. In particular, if p is odd, then typically no automorphism like the φin of Definition 4 exists since such an automorphism must necessarily haveeven order. On the other side of the coin however, most of their results giveexamples with large numbers of generators and examples that are 2-generatedseem difficult to construct.4. Purely non-strongly Beauville groupsWhat about purely non-strongly real Beauville groups? In [4, Lemma 2.2]the author shows, in the terminology defined here, that the Mathieu groupsM11 and M23 are purely non-strongly real Beauville groups, indeed the realcontent of [5, Conjecture 1] is that among the non-abelian finite simple groupsthese are really the only ones. For an infinite supply of examples we have thefollowing.Proposition 14. If G and H are Beauville groups of coprime order, such thatG is a purely non-strongly real Beauville group, then G×H is a purely non-strongly real Beauville group.Proof. Let {(x1, y1), (x2, y2)} be a Beauville structure forG and let {(u1, v1), (u2, v2)}be a Beauville structure for H. Since |G| and |H| are coprime it follows that{((x1, u1), (y1, v1)), ((x2, u2), (y2, v2))}is a Beauville structure for G × H. Since |G| is coprime to |H| it followsthat G and H are not isomorphic and so Aut(G × H)=Aut(G)×Aut(H).Since elements of Aut(G) cannot be used to provide a strongly real Beauvillestructure, it follows that none of the Beauville structures of G×H are stronglyreal. Corollary 15. There exist infinitely many purely non-strongly real Beauvillegroups.Purely (Non-)Strongly Real Beauville Groups 7Proof. As noted above, M11 is a purely non-strongly real Beauville group. InProposition 14 we can therefore takeG to be M11. Since |M11| = 11×5×32×24we can take H to be any Beauville group has order coprime to 11, 5, 3 and 2.Infinitely many examples of such groups are constructed in [1, 7, 10, 11, 14]as well as in Proposition 11. The next example shows that Proposition 14 is far from being the bestpossible.Example 16. The group M11 × A5 is easily seen to be a Beauville group asthe elementsx1 := (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11)(12, 13, 14, 15, 16),y1 := (1, 5, 3, 4, 10, 2, 8, 9, 11, 6, 7)(12, 14, 15, 13, 16)along withx2 := (1, 2, 9, 10, 6)(3, 11, 5, 4, 7)(12, 13, 14, 15, 16),y2 := (1, 4, 8, 11, 3)(2, 9, 7, 5, 6)(12, 14, 15, 13, 16)provide a Beauville structure of type ((55,55,55),(5,5,5)). The lack of auto-morphisms that make M11 a purely non-strongly real Beauville group clearlyalso make M11 × A5 a purely non-strongly real Beauville group. Note thatevery prime dividing the order A5 also divides the order of M11 and that A5not even a Beauville group.Question 17. What is the most general form of Proposition 14?The author is not aware of any nilpotent or even soluble examples. Aspreviously mentioned ‘the automorphism group of a p-group is typically a p-group’. Among the groups of order pn for small n few examples of 2-generatedgroups with an automorphism group of odd order exist and the few that doappear to not be Beauville groups, suggesting that such examples are actuallyquite difficult to construct.Question 18. Do there exist any nilpotent or soluble purely non-strongly realBeauville groups?This would be immediately answered by combining Proposition 14 withan answer to the following.Question 19. Do there exist Beauville p-groups whose automorphism groupshave odd order?References[1] N. W. Barker, N. Bosten and B. T. Fairbairn “A note on Beauvillep-groups” Exp. Math., 21(3): 298–306 (2012) arXiv:1111.3522v2doi:10.1080/10586458.2012.669267[2] I. Bauer, F. Catanese and F. Grunewald, Beauville surfaces without real struc-tures, in Geometric methods in algebra and number theory pp. 1–42, Progr.Math., 235, Birkhuser Boston, Boston, MA, 2005.8 Ben Fairbairn[3] F. Catanese, Fibered surfaces, varieties isogenous to a product and relatedmoduli spaces, Amer. J. Math. 122 (2000), no. 1, pp. 1–44[4] B. T. Fairbairn “Some Exceptional Beauville Structures” J. Group Theory,15(5), pp. 631–639 (2012) arXiv:1007.5050 DOI: 10.1515/jgt-2012-0018[5] B. T. Fairbairn, “Strongly Real Beauville Groups” in Beauville Surfaces andGroups, Springer Proceedings in Mathematics & Statistics, Vol. 123 (eds I.Bauer, S. Garion and A. Vdovina), Springer-Verlag (2015) pp. 41–61[6] B. T. Fairbairn, More on Strongly Real Beauville Groups, in Symmetries inGraphs Maps and Polytopes 5th SIGMAP Workshop, West Malvern, UK, July2014 (eds. J. Sˇira´nˇ and R. Jajcay) Springer Proceedings in Mathematics &Statistics 159 (2016) pp. 129–146[7] B. T. Fairbairn, A new infinite family of non-abelian strongly real Beauvillep-groups for every odd prime p, Bull. Lond. Math. Soc. 49(4) (2017)doi:10.1112/blms.12060 arXiv:1608.00774[8] Y. Fuertes and G. Gonza´lez-Diez “On Beauville structures on the groups Snand An” Math. Z. 264 (2010), no. 4, 959–968[9] Y. Fuertes and G. Jones “Beauville surfaces and finite groups” J. Algebra 340(2011) 13–27[10] S¸. Gu¨l, A note on strongly real Beauville p-groups, Monatsh. Math. (2017)doi:10.1007/s00605-017-1034-1 arXiv:1607.08907[11] S¸. Gu¨l, An infinite family of strongly real Beauville p-groups, preprint 2016arXiv:1610.06080[12] G. T. Helleloid and U. Martin, The automorphism group of a finite p-group isalmost always a p-group, J. Algebra 312 (2007) pp. 284–329[13] A. M. MacBeath “Generators of the linear fractional groups” Number Theory(Proc. Sympos. Pure Math., Vol. XII, Houston, Tex., 1967), Amer. Math. Soc.,Providence, R.I., 1969, pp. 14–32[14] J. Stix and A. Vdovina, Series of p-groups with Beauville structure,Monatsh. Math. 181 (2016), no. 1, pp. 177–186 doi:10.1007/s00605-015-0805-9arXiv:1405.3872Ben FairbairnDepartment of Economics, Mathematics and StatisticsBirkbeck, University of LondonMalet StLondonWC1E 7HXUnited Kingdome-mail: b.fairbairn@bbk.ac.uk

English

Purely (Non-)StronglyReal Beauville GroupsByBen FairbairnBirkbeck Pure Mathematics Preprint Series Preprint Number 48www.bbk.ac.uk/ems/research/pure/preprintsPURELY (NON-)STRONGLY REAL BEAUVILLE GROUPSBEN FAIRBAIRNAbstract. We discuss Beauville groups whose corresponding Beauville sur-faces are either always strongly real or never strongly real producing severalinfinite families of examples.1. IntroductionDefinition 1.1. A surface S is a Beauville surface if• the surface S is isogenous to a higher product, that is, S ∼= (C1 × C2)/Gwhere C1 and C2 are algebraic curves of genus at least 2 and G is a finitegroup acting faithfully on C1 and C2 by holomorphic transformations in sucha way that it acts freely on the product C1 × C2, and• each Ci/G is isomorphic to the projective line P1(C) and the covering mapCi → Ci/G is ramified over three points.These surfaces were first defined by Catanese in [3] and the first significantinvestigation of them was conducted by Bauer, Catanese and Grunewald in [2].They have numerous nice properties and are relatively easy to construct makingthem useful for producing counterexamples and testing conjectures. The followingcondition is also investigated in [2].Definition 1.2. Let S be a complex surface. We say that S is strongly real ifthere exists a biholomorphism σ : S → S such that σ ◦ σ is the identity map.What makes these surfaces particularly easy to work with is that all of the abovecan be easily translated into group theoretic terms.Definition 1.3. Let G be a finite group. Let x, y ∈ G and letΣ(x, y) :=|G|⋃i=1⋃g∈G{(xi)g, (yi)g, ((xy)i)g}.A Beauville structure for the group G is a set of pairs of elements {(x1, y1), (x2, y2)} ⊂G×G with the property that 〈x1, y1〉 = 〈x2, y2〉 = G such thatΣ(x1, y1) ∩ Σ(x2, y2) = {e}.If G has a Beauville structure we say that G is a Beauville group.A group defines a Beauville surface if and only if it has a Beauville structure.Furthermore, the Beauville surface defined by a particular Beauville structure isstrongly real if and only if the corresponding Beauville structure has the propertyof being strongly real that we define as follows.Definition 1.4. Let G be a Beauville group and let X = {(x1, y1), (x2, y2)} be aBeauville structure for G. We say that G and X are strongly real if there existsan automorphism φ ∈ Aut(G) and elements gi ∈ G for i = 1, 2 such thatgiφ(xi)g−1i = x−1i and giφ(yi)g−1i = y−1i .1Most of the Beauville structures appearing in the literature are either explicitlyshown to be strongly real or the question of reality is never pursued. In manygroups there are elements that can never be inverted by automorphisms meaningany Beauville structure defined using such elements cannot be strongly real, indeedgroups typically have numerous Beauville structures some of which are stronglyreal, some of which are not. Here we are interested in the extreme cases and thusmake the following definition.Definition 1.5. A finite group G is a purely strongly real Beauville group ifG is a Beauville group such that every Beauville structure of G is strongly real. Afinite group G is a purely non-strongly real Beauville group if G is a Beauvillegroup such that none of its Beauville structures are strongly real.Throughout we shall follow the conventions that in a group G and elementsg, h ∈ G we have that gh = hgh−1 and [g, h] = ghg−1h−1.This paper is organised as follows. In Section 2 we will give infinitely manyexamples of Beauville groups that have strongly real Beauville structures as well asBeauville structures that are not suggesting that groups typically lie in neither ofthe categories in Definition 1.5. Despite this, we go on in Section 3 to give infinitelymany examples of purely strongly real Beauville groups before in the final sectiongiving infinitely many examples of purely non-strongly real Beauville groups.2. Neither CaseWe first prove results showing that Beauville groups typically fall into neithercase with several different examples.Lemma 2.1. If n > 5, then the alternating group An is neither a Purely stronglyreal Beauville group nor a non-strongly real Beauville group.Proof. Strongly real Beauville structures for these groups were constructed byFuertes and Gonza´lez-Diez in [8] so it is sufficient to construct non-strongly realBeauville structures for these groups. We consider the cases n odd and n even sep-arately. Recall that in either case, if two permutations have different cycle type,then they cannot be conjugate.First suppose that n ≥ 7 is odd. Letx1 := (1, 2, 4) and y1 := (1, 2, 3, 4, . . . , n).We have that their product is the n-cycle x1y1 = (1, 3, 4, 2, 5, . . . , n) by direct calcu-lation. By considering the subgroup generated by the elements xyi11 we have that thesubgroup these elements generate is 2-transitive and therefore primitive. Since thissubgroup contains the 3-cycle x1 it follows that these elements generate the wholeof the alternating group. It is easy to see that no automorphism simultaneouslyinverts both of these elements so any permutations that can be used to extend thisto a Beauville structure gives a non-strongly real Beauville structure. Consider thepermutationsx2 := (5, 4, 3, 2, 1) and y2 := (3, 4, 5, 6, . . . , n).We have that their product is the (n − 2)-cycle x2y2 = (1, 6, . . . , n, 3, 2) by directcalculation. By considering the subgroup generated by the elements xyi22 we have thatthe subgroup these elements generate is 2-transitive and therefore primitive. Sincethis subgroup contains the double-transposition [x2, y2] = (1, 5)(3, n) it follows thatthese elements generate the whole of the alternating group. We therefore have anon-strongly real Beauville group.Next, suppose n ≥ 8 is even. Letx1 := (1, 2)(3, 4) and y1 := (2, 3, 4, . . . , n).2We have that their product is the (n − 1)-cycle x1y1 = (1, 3, 5, . . . , n, 2) by directcalculation. By considering the subgroup generated by the elements xyi11 we have thatthe subgroup these elements generate is 2-transitive and therefore primitive. Sincethis subgroup contains the double-transposition x1 it follows that these elementsgenerate the whole of the alternating group. It is easy to see that no automorphismsimultaneously inverts both of these elements so any permutations that can be usedto extend this to a Beauville structure gives a non-strongly real Beauville structure.Finally, letx2 := (1, 2, 3)(4, 5, . . . , n) and y2 := (5, 4, 3, 2, 1).We have that their product is the (n − 3)-cycle x2y2 = (3, 5, . . . , n). By consider-ing the subgroup generated by the elements yxi22 we have that the subgroup theseelements generate is 2-transitive and therefore primitive. Since this subgroup con-tains the double transposition [x2, y22 ] = (2, 5)(3, n) it follows that these elementsgenerate the whole of the alternating group. We therefore have a non-strongly realBeauville group.The case of A6 is complicated by the existence of exceptional outer automor-phisms but despite this can easily be handled separately. Further examples are given by the following.Lemma 2.2. None of the Suzuki groups 2B2(22n+1) are purely (non-)strongly realBeauville groups.Proof. The Beauville structures constructed by Fuertes and Jones in [9, Theorem6.2] are non-strongly real since they take y1 as having order 4 and no automor-phism maps such elements to their inverses in these groups. Strongly real Beauvillestructures for these groups were constructed by the author in [5]. Lemma 2.3. Aside from the Mathieu groups M11 and M23 none of the sporadicsimple groups are purely (non-)strongly real Beauville groups.Proof. Aside from M11 and M23, strongly real Beauville structures for these groupswere constructed by the author in [4]. Non-strongly real Beauville structures areeasily obtained computationally and with character theory, the more difficult casesbeing easily dealt with thanks to the existence of elements that cannot be sentto their inverses by any automorphism at all such as elements of order 71 in themonster group M and elements of order 47 in the baby monster group B. We will return to the cases of M11 and M23 in Section 4 since they are genuineexceptions to the above.It is clear that numerous further examples can be constructed from the aboveusing direct products.3. Purely Strongly Real Beauville GroupsFirstly, the following observation has been made elsewhere in the literature manytimes and provides infinitely many purely strongly real Beauville groups.Lemma 3.1. A finite abelian group is a strongly real Beauville group if and onlyif it is a purely strongly real Beaville group.Proof. In any abelian group the homomorphism x 7→ −x inverts every element. Abelian Beauville groups were first constructed by Catanese in [3] and classifiedby Bauer, Catanese and Grunewald in [2]. For non-soluble examples, we have thefollowing infinite supply.3Proposition 3.2. Let q > 2 be a power of 2 and let k be a positive integer. If(q, k) 6= (4, 1), then the characteristically simple group L2(q)k is a purely stronglyreal Beaville group whenever it is 2-generated.Proof. In [13] MacBeath observed that any generating pair for the groups L2(q)can be inverted by an inner automorphism when q is even and since these groupshave only one class of involutions, the only elements of even order. We remark that the exception (q, k) = (4, 1) is a genuine exception thanks to theisomorphism A5 ∼= L2(4) and this group is well known to not be a Beauville group.For each prime p ≥ 5 we can also construct infinitely many (new) non-abeliannilpotent examples as follows.Proposition 3.3. If p ≥ 5 is a prime and n ≥ r ≥ 1 are integers, then the groupG := 〈x, y, z |xpn , ypn , zpr [x, y] = z, [x, z], [y, z]〉is a purely strongly real Beauville group.Proof. Let {(x1, y1), (x2, y2)} be a Beauville structure of G. We first note thatAut(G) acts transitively on the non-central elements of G of a given order and agenerating pairs must consist of two element of order pn, so without loss we mayassume that x1 = x. Similarly CAut(G)(x) acts transitively the elements x cangenerate with so without loss we may also assume that y1 = y and that φ ∈Aut(G)acts by xφ = x−1 and yφ = y−1 (so we can take the element g1 of Definition 1.4 tobe trivial).Observe the following. For any element g ∈ G\Φ(G) we have that its conjugatesare gG = {gzi | i = 0, . . . , pr − 1}. Every element of this group can be written inthe form xiyjzk for some 0 ≤ i, j ≤ pn − 1 and 0 ≤ k ≤ pr − 1, so we havethat x2 = xi1yj1zk1 and y2 = xi2yj2zk2 for some 1 ≤ i1, i2, j1, j2 ≤ pn − 1 and0 ≤ k1, k2 ≤ pr − 1. Notice that if these are chosen so that i1 6= j1, i2 6= j2,i1 + i2 6= j1 + j2 and o(x2) = o(y2) = pn, then these elements provide a Beauvillestructure and since p ≥ 5 finding such integers is straightforward. Moreover(xiyjzk)φ = x−iy−jzk and (xiyjzk)−1 = x−iy−jz−k−ijWe also have that for any 0 ≤ a, b ≤ pn − 1(x−iy−jzk)xayb = x−iy−jzk+bi−aj .It follows that to have (xφ2 )xayb = x−12 and (yφ2 )xayb = y−12 we must have thatbi1 − aj1 ≡ −2k1 − ij (mod pr) and bi2 − aj2 ≡ −2k2 − ij (mod pr).If the values of i1, i2, j1, j2, k1 and k2 are chosen so x2 and y2 generate the groupand the conjugacy condition is satisfied, then values of a and b satisfying theseequations can be found if j2i1 6≡ j1i2 (mod p). We claim that if j2i1 ≡ j1i2 (modp), then x2 and y2 do not generate the group. Note that under this condition wehave that(xi1yj1)i2 = xi1i2yj1i2zk = xi1i2yj2i1zkfor some k but we also have that(xi2yj2)i1 = xi1i2yj2i1zk′for some k′ in other words y2 ∈ 〈x2, z〉 which is a proper subgroup. We remark that strongly real Beauville p-groups have proved somewhat difficultto construct the only previously known examples being given in [6, 7, 10, 11]. Theabove provides infinitely many further new examples and does so for each primep ≥ 5.4Clearly Proposition 3.2, Lemma 3.1 and Proposition 3.3 can be combined toproduce infinitely many other examples like L2(8) × C25 but it is not clear whatother examples can arise.Problem 3.4. Find other examples of purely strongly real Beauville groups.In particular, we have the following question.Question 3.5. Do there exist purely strongly real Beauville 2-groups and 3-groups.In the opinion of the author it seems likely that 2-generated 2-groups are morelikely to be purely strongly real Beauville groups: there is a general philosophyin the study of p-groups that ‘the automorphism group of a p-group is typicallya p-group’ thanks to the results of Helleloid and Martin [12]. In particular, if p isodd, then typically no automorphism like the φ in of Definition 1.4 exists since suchan automorphism must necessarily have even order. On the other side of the coinhowever, most of their results give examples with large numbers of generators andexamples that are 2-generated seem difficult to construct.4. Purely non-strongly Beauville groupsWhat about purely non-strongly real Beauville groups? In [4, Lemma 2.2] theauthor shows, in the terminology defined here, that the Mathieu groups M11 andM23 are purely non-strongly real Beauville groups, indeed the real content of [5,Conjecture 1] is that among the non-abelian finite simple groups these are reallythe only ones. For an infinite supply of examples we have the following.Proposition 4.1. If G and H are Beauville groups of coprime order, such that Gis a purely non-strongly real Beauville group, then G ×H is a purely non-stronglyreal Beauville group.Proof. Let {(x1, y1), (x2, y2)} be a Beauville structure forG and let {(u1, v1), (u2, v2)}be a Beauville structure for H. Since |G| and |H| are coprime it follows that{((x1, u1), (y1, v1)), ((x2, u2), (y2, v2))}is a Beauville structure for G × H. Since |G| is coprime to |H| it follows that Gand H are not isomorphic and so Aut(G × H)=Aut(G)×Aut(H). Since elementsof Aut(G) cannot be used to provide a strongly real Beauville structure, it followsthat none of the Beauville structures of G×H are strongly real. Corollary 4.2. There exist infinitely many purely non-strongly real Beauville groups.Proof. As noted above, M11 is a purely non-strongly real Beauville group. In Propo-sition 4.1 we can therefore take G to be M11. Since |M11| = 11× 5× 32× 24 we cantake H to be any Beauville group has order coprime to 11, 5, 3 and 2. Infinitelymany examples of such groups are constructed in [1, 7, 10, 11, 14] as well as inProposition 3.3. The next example shows that Proposition 4.1 is far from being the best possible.Example 4.3. The group M11 × A5 is easily seen to be a Beauville group as theelementsx1 := (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11)(12, 13, 14, 15, 16),y1 := (1, 5, 3, 4, 10, 2, 8, 9, 11, 6, 7)(12, 14, 15, 13, 16)along withx2 := (1, 2, 9, 10, 6)(3, 11, 5, 4, 7)(12, 13, 14, 15, 16),y2 := (1, 4, 8, 11, 3)(2, 9, 7, 5, 6)(12, 14, 15, 13, 16)5provide a Beauville structure of type ((55,55,55),(5,5,5)). The lack of automor-phisms that make M11 a purely non-strongly real Beauville group clearly also makeM11×A5 a purely non-strongly real Beauville group. Note that every prime dividingthe order A5 also divides the order of M11 and that A5 not even a Beauville group.Question 4.4. What is the most general form of Proposition 4.1?The author is not aware of any nilpotent or even soluble examples. As previouslymentioned ‘the automorphism group of a p-group is typically a p-group’. Amongthe groups of order pn for small n few examples of 2-generated groups with an auto-morphism group of odd order exist and the few that do appear to not be Beauvillegroups, suggesting that such examples are actually quite difficult to construct.Question 4.5. Do there exist any nilpotent or soluble purely non-strongly realBeauville groups?This would be immediately answered by combining Proposition 4.1 with an an-swer to the following.Question 4.6. Do there exist Beauville p-groups whose automorphism groups haveodd order?References[1] N. W. Barker, N. Bosten and B. T. Fairbairn “A note on Beauville p-groups” Exp. Math.,21(3): 298–306 (2012) arXiv:1111.3522v2 doi:10.1080/10586458.2012.669267[2] I. Bauer, F. Catanese and F. Grunewald, Beauville surfaces without real structures, in Geo-metric methods in algebra and number theory pp. 1–42, Progr. Math., 235, Birkha¨user Boston,Boston, MA, 2005.[3] F. Catanese, Fibered surfaces, varieties isogenous to a product and related moduli spaces,Amer. J. Math. 122 (2000), no. 1, pp. 1–44[4] B. T. Fairbairn “Some Exceptional Beauville Structures” J. Group Theory, 15(5), pp. 631–639(2012) arXiv:1007.5050 DOI: 10.1515/jgt-2012-0018[5] B. T. Fairbairn, “Strongly Real Beauville Groups” in Beauville Surfaces and Groups, SpringerProceedings in Mathematics & Statistics, Vol. 123 (eds I. Bauer, S. Garion and A. Vdovina),Springer-Verlag (2015) pp. 41–61[6] B. T. Fairbairn, More on Strongly Real Beauville Groups, in Symmetries in Graphs Mapsand Polytopes 5th SIGMAP Workshop, West Malvern, UK, July 2014 (eds. J. Sˇira´nˇ and R.Jajcay) Springer Proceedings in Mathematics & Statistics 159 (2016) pp. 129–146[7] B. T. Fairbairn, A new infinite family of non-abelian strongly real Beauville p-groups for everyodd prime p, Bull. Lond. Math. Soc. 49(4) (2017) doi:10.1112/blms.12060 arXiv:1608.00774[8] Y. Fuertes and G. Gonza´lez-Diez “On Beauville structures on the groups Sn and An” Math.Z. 264 (2010), no. 4, 959–968[9] Y. Fuertes and G. Jones “Beauville surfaces and finite groups” J. Algebra 340 (2011) 13–27[10] S¸. Gu¨l, A note on strongly real Beauville p-groups, Monatsh. Math. (2017)doi:10.1007/s00605-017-1034-1 arXiv:1607.08907[11] S¸. Gu¨l, An infinite family of strongly real Beauville p-groups, preprint 2016arXiv:1610.06080[12] G. T. Helleloid and U. Martin, The automorphism group of a finite p-group is almost alwaysa p-group, J. Algebra 312 (2007) pp. 284–329[13] A. M. MacBeath “Generators of the linear fractional groups” Number Theory (Proc. Sympos.Pure Math., Vol. XII, Houston, Tex., 1967), Amer. Math. Soc., Providence, R.I., 1969, pp.14–32[14] J. Stix and A. Vdovina, Series of p-groups with Beauville structure, Monatsh. Math. 181(2016), no. 1, pp. 177–186 doi:10.1007/s00605-015-0805-9 arXiv:1405.3872Ben Fairbairn, Department of Economics, Mathematics and Statistics, Birkbeck, Uni-versity of London, Malet Street, London WC1E 7HX, United KingdomE-mail address: b.fairbairn@bbk.ac.uk6

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We discuss Beauville groups whose corresponding Beauville surfaces\ud
are either always strongly real or never strongly real producing several\ud
infinite families of examples

Purely (Non-)Strongly Real Beauville Groups

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