Simple groups, product actions, and generalised quadrangles
aa r X i v : . [ m a t h . G R ] F e b Simple groups, product actions, and generalised quadrangles
John Bamberg, Tomasz Popiel, Cheryl E. Praeger
Abstract.
The classification of flag-transitive generalised quadrangles is a long-standing open problemat the interface of finite geometry and permutation group theory. Given that all known flag-transitivegeneralised quadrangles are also point-primitive (up to point–line duality), it is likewise natural to seeka classification of the point-primitive examples. Working towards this aim, we are led to investigategeneralised quadrangles that admit a collineation group G preserving a Cartesian product decomposi-tion of the set of points. It is shown that, under a generic assumption on G , the number of factors ofsuch a Cartesian product can be at most four. This result is then used to treat various types of primitiveand quasiprimitive point actions. In particular, it is shown that G cannot have holomorph compound O’Nan–Scott type. Our arguments also pose purely group-theoretic questions about conjugacy classesin non-Abelian finite simple groups, and about fixities of primitive permutation groups.
1. Introduction
Generalised quadrangles are point–line incidence geometries introduced by Tits [ ] in an attemptto find geometric models for simple groups of Lie type. The classical generalised quadrangles arisein this way [ , Section 3]. Each admits one of the simple classical groups T = PSp(4 , q ) ∼ = Ω ( q ),PSU(4 , q ) ∼ = PΩ − ( q ) or PSU(5 , q ) acting transitively on flags (incident point–line pairs). Moreover, thepoint and line stabilisers are certain maximal subgroups of T , so T acts primitively on both points andlines. The classification of flag-transitive generalised quadrangles is a long-standing open problem. Inaddition to the classical families, only two other flag-transitive examples are known (up to point–lineduality), each admitting an affine group acting point-primitively but line-imprimitively. Hence, all ofthe known flag-transitive generalised quadrangles are also point-primitive (up to duality), and so it isnatural to seek a classification of the point-primitive examples. Indeed, this is arguably a more difficultproblem, because one begins with essentially no information about the action of the collineation groupon lines, nor any notion of what ‘incidence’ means, whereas in a flag-transitive point–line geometry,points and lines correspond to cosets of certain subgroups of the collineation group, and incidence isdetermined by non-empty intersection of these cosets.Here we prove the following theorem. The abbreviations HS (holomorph simple), HC (holomorphcompound), SD (simple diagonal), CD (compound diagonal), PA (product action), AS (almost simple)and TW (twisted wreath) refer to the possible types of non-affine primitive permutation group actions,in the sense of the O’Nan–Scott Theorem as stated in [ , Section 6]. In the second column ofTable 1, soc( G ) denotes the socle of the group G , namely the subgroup generated by its minimalnormal subgroups. By fix Ω ( h ) we mean the number of elements fixed by a permutation h of the setΩ, and Q − (5 ,
2) is the unique generalised quadrangle of order (2 , Theorem . If Q is a thick finite generalised quadrangle with a non-affine collineation group G that acts primitively on the point set P of Q , then the action of G on P does not have O’Nan–Scotttype HC, and the conditions in Table 1 hold for the remaining O’Nan–Scott types. Mathematics Subject Classification. primary 51E12; secondary 20B15, 05B25.
Key words and phrases. generalised quadrangle, primitive permutation group, finite simple group, centraliser, fixity.The first author acknowledges the support of the Australian Research Council (ARC) Future FellowshipFT120100036. The second author acknowledges the support of the ARC Discovery Grant DP140100416. The researchreported in the paper forms part of the ARC Discovery Grant DP140100416 of the third author. We thank Elisa Covatoand Tim Burness for making available to us the results quoted in Section 6, and Luke Morgan for helpful discussions.
Type soc( G ) Necessary conditionsHS T × T T is of Lie type A ε , A ε , B , C , C , D ε , D ε , D ε , E ε , E or F SD T k T is a sporadic simple group or T ∼ = Alt n with n
18; or T is an exceptional Lie type group; or T has Lie type A , A εn with 2 n
8; B n or C n with 2 n
4; or D εn with 4 n T k ) r r = 2 and T ∼ = Alt n with n
9; or r = 2 and T is a sporadic simple group with T = Suz, Co , Fi , Fi , B or M; or r = 2 and T has Lie type A , A ε , A ε , B , B , F , G or G ; or r = 3 and T ∼ = J or T is of Lie type A or B PA T r r = 2 and fix Ω ( h ) < | Ω | / for all h ∈ H \ { } ; or3 r T is a group of Lie type, and fix Ω ( h ) < | Ω | − r/ for all h ∈ H \ { } ; or3 r H = T ∼ = Alt p with point stabiliser p. p − for a prime p ≡ T fix P ( g ) < |P| / for all g ∈ G \ { } ; or Q = Q − (5 ,
2) with T ∼ = PSU (2)TW T r fix P ( g ) < |P| / for all g ∈ G \ { } Table 1.
Conditions for Theorem 1.1. Here T is a non-Abelian finite simple group, k > r >
2. If G acts primitively of type CD (respectively PA) on P , then G H ≀ Sym r forsome primitive group H Sym(Ω) of type SD (respectively AS) with socle T k (respectively T ). Note also that, in the notation for finite simple groups of Lie type used in Table 1 (and throughoutthe paper), ε = ± and A + n = A n , A − n = A n , D + n = D n , D − n = D n , E +6 = E , E − = E .Before we proceed, a remark is in order about the assumption in Theorem 1.1 that G not bean affine group. If G is affine, then the generalised quadrangle Q necessarily arises from a so-called pseudo-hyperoval in a projective space PG(3 n − , q ) with q even [ ]. In joint work with Glasby [ ],we were able to classify the generalised quadrangles admitting an affine group that acts primitivelyon points and transitively on lines: they are precisely the two flag-transitive, point-primitive, line-imprimitive generalised quadrangles mentioned above. However, without the extra assumption oftransitivity on lines, the problem is equivalent to the classification of the pseudo-hyperovals that havean irreducible stabiliser. As explained in [ , Remark 1.3], this latter problem would appear to beextremely difficult, and possibly intractable. It also has a rather different flavour to the cases treatedin the present paper, and so we do not consider it further here.Let us now establish some definitions and notation, before discussing further. By a point–lineincidence geometry we mean a triple Γ = ( P , L , I ), where P and L are sets whose elements are called points and lines , respectively, and I ⊆ P × L is a symmetric binary relation called incidence . We writeΓ = ( P , L ) instead of ( P , L , I ) when we do not need to refer to the incidence relation explicitly. Twopoints (respectively lines) of Γ are said to be collinear (respectively concurrent ) if they are incidentwith a common line (respectively point). A collineation of Γ is a permutation of
P ∪ L that preserves P and L setwise and preserves the incidence relation. By a collineation group of Γ we mean a subgroupof the group of all collineations of Γ, which is called the full collineation group .A generalised quadrangle is a point–line incidence geometry Q = ( P , L ) that satisfies the followingtwo axioms: (i) two distinct points are incident with at most one common line, and (ii) given a point P and a line ℓ not incident with P , there exists a unique point incident with ℓ that is collinear with P .The second axiom implies that every pair in P ∪ L is contained in an ordinary quadrangle, and that Q contains no triangles. All generalised quadrangles considered in this paper are assumed to be finite,in the sense that P and L are finite sets. If every point is incident with at least three lines, and everyline is incident with at least three points, then Q is said to be thick . In this case, there exist constants s > t > t + 1 lines and every line is incident withexactly s + 1 points [ , Corollary 1.5.3]. The pair ( s, t ) is called the order of Q . Observe also thatthere is a natural concept of point–line duality for generalised quadrangles: if ( P , L ) is a generalisedquadrangle, then so is ( L , P ); and if ( P , L ) has order ( s, t ), then ( L , P ) has order ( t, s ). IMPLE GROUPS, PRODUCT ACTIONS, AND GENERALISED QUADRANGLES 3
Let us now discuss Theorem 1.1 further. The primitive permutation groups on a finite set ∆ areclassified into eight types according to the O’Nan–Scott Theorem as presented in [ , Section 6]. In2012, Bamberg et al. [ ] showed that if a thick finite generalised quadrangle admits a collineationgroup G that acts primitively on both points and lines, then G must be an almost simple (AS type)group. That is, T G Aut( T ) for some non-Abelian finite simple group T . Given that thereexist point-primitive generalised quadrangles that are line-transitive but line-imprimitive, our initialaim was to extend the result of [ ] by relaxing the line-primitivity assumption to line-transitivity. Inaddition to handling the affine (HA type) case with Glasby [ ], we were also able to show that no suchexamples arise if the point action has type HS or HC [ ].Theorem 1.1 significantly strengthens and expands upon the results of [
3, 8 ]. The idea behind itsproof begins with the following observations. A primitive group G Sym(∆) of O’Nan–Scott type HC,CD, PA or TW preserves a Cartesian product decomposition ∆ = Ω r , for some set Ω and some r > semiregular permutationgroup action is one in which only the identity element fixes a point, and if H , . . . , H r are permutationgroups on sets Ω , . . . , Ω r , respectively, then the product action of the direct product Q ri =1 H i on theCartesian product Q ri =1 Ω i is the action ( ω , . . . , ω r ) ( h ,...,h r ) = ( ω h , . . . , ω h r r ). We also recall that apermutation group is said to act regularly if it acts transitively and semiregularly. Theorem . Let Ω , . . . , Ω r be finite sets with | Ω | · · · | Ω r | , where r > , and let H i Sym(Ω i ) for each i ∈ { , . . . , r } . Assume further that H is non-trivial and that its action on Ω is not semiregular. Suppose that N = Q ri =1 H i is a collineation group of a thick finite generalisedquadrangle Q = ( P , L ) of order not equal to (2 , , such that P = Q ri =1 Ω i and N has the productaction on P . Then r , and every non-identity element of H fixes less than | Ω | − r/ points of Ω . The proof of Theorem 1.2 relies on the existence of a non-identity element h of H that fixes atleast one point of Ω . If r >
2, one can then construct a collineation ( h , , . . . , ∈ N of Q thatfixes at least Q ri =2 | Ω i | points of the Cartesian product P = Q ri =1 Ω i . Theorem 1.2 is then deducedfrom the following result, which bounds the number of points fixed by a non-identity collineation ofan arbitrary thick finite generalised quadrangle. The proofs of both theorems are given in Section 2. Theorem . Let θ be a non-identity collineation of a thick finite generalised quadrangle Q =( P , L ) . Then either θ fixes less than |P| / points of Q , or Q is the unique generalised quadrangle Q − (5 , of order (2 , and θ fixes exactly of the points of Q . Remark . Theorem 1.3 improves a particular case of a recent result of Babai on automorphismgroups of strongly regular graphs [ , Theorem 1.7]. If Q has order ( s, t ) then its collinearity graph,namely the graph with vertex set P and two vertices adjacent if and only if they are collinear in Q ,is a strongly regular graph with parameters v = |P| = ( s + 1)( st + 1), k = s ( t + 1), λ = s − µ = t + 1. Roughly speaking, we have v ≈ s t and k ≈ st , so the condition k n / in assertion (b)of [ , Theorem 1.7] becomes t s , which is just Higman’s inequality for generalised quadrangles (seeLemma 2.1(ii)). Babai’s result, which applies far more generally to strongly regular graphs that are non-trivial , non-graphic and non-geometric , therefore implies that a non-identity collineation θ of Q can fix at most O ( |P| / ) points. Theorem 1.3 sharpens the 7 / / , Theorem 1.7]sharpens the 7 / / t > s : the condition k > n / roughly translates to t > s , and the corresponding bound is O ( √ kn ), with √ kn ≈ s / t > ( s t ) / ≈ |P| / when t > s .)To aid our discussion, let us now state the following immediate corollary of Theorem 1.2. Corollary . Let Ω be a finite set with | Ω | > , and suppose that H Sym(Ω) is non-trivialand not semiregular. Suppose that N = H r , r > , is a collineation group of a thick finite generalisedquadrangle Q = ( P , L ) of order not equal to (2 , , such that P = Ω r and N has the product action on P . Then r , and every non-identity element of H fixes less than | Ω | − r/ points of Ω . JOHN BAMBERG, TOMASZ POPIEL, CHERYL E. PRAEGER
T r = 1 r = 2 r = 3Alt n n
18 5 n n , Fi , Fi , B, M J exceptional Lie type any F ( q ), G ( q ), G ( q ), B ( q ) B ( q )PSL n +1 ( q ) 1 n n n = 1, q = 7PSU n +1 ( q ) 2 n n n ( q ) or Ω n +1 ( q ) 2 n n = 2 —PΩ ± n ( q ) 4 n Table 2.
Possibilities for a non-Abelian finite simple group T with the property that | C T ( x ) | < | T | − r/ for all x ∈ T \ { } , for r equal to one of 1, 2 or 3. We apply Corollary 1.5 to groups N that arise as subgroups of certain types of primitive groups.This in turn motivates certain questions about non-Abelian finite simple groups. As illustration,consider the case where Ω = T for some non-Abelian finite simple group T , with H = T × T actingon Ω via ω ( x,x ′ ) = x − ωx ′ . This situation arises when N is the socle (the subgroup generated by theminimal normal subgroups) of a primitive group of type HS ( r = 1) or HC ( r > x ′ = x thenthe element ( x, x ′ ) = ( x, x ) ∈ H fixes precisely | C T ( x ) | points of Ω, where C T ( x ) is the centraliser of x in T . Corollary 1.5 therefore implies that r
4, and that | C T ( x ) | < | T | − r/ for all x ∈ T \ { } .We therefore ask which non-Abelian finite simple groups T satisfy this condition. If r = 4 then werequire that | C T ( x ) | < | T | / for all x ∈ T \ { } , which is false for every non-Abelian finite simplegroup T . Indeed, it is well known that every non-Abelian finite simple group T contains an involution x with | C T ( x ) | > | T | / (in fact, every involution in T has this property [ , Proposition 2.4]). For r ∈ { , , } , we verify the following result in Section 3. Although this result follows from routinecalculations, we include it here in case it proves to be a convenient reference. Proposition . Let r ∈ { , , } and let T be a non-Abelian finite simple group. Then either | C T ( x ) | > | T | − r/ for some x ∈ T \ { } , or T is one of the groups listed in Table 2. Our new results about generalised quadrangles with point-primitive collineation groups are provedin Sections 4–6. Corollary 1.5 is applied not only to actions of type HS or HC as illustrated above, butalso to types SD, CD and PA. In particular, the proof of Theorem 1.1 is free from the Classificationof Finite Simple Groups (CFSG) to the extent that, for G of type HC, CD or PA with socle T r × T r , T ( k − r or T r respectively, the proof that r H inCorollary 1.5 is an almost simple primitive group, so we are led to consider lower bounds on the fixity of such a group, namely the maximum number of fixed points of a non-identity element. In Section 6,we discuss how refinements of a recent result of Liebeck and Shalev [ , Theorem 4] on this problem,currently being carried out by Elisa Covato at the University of Bristol as part of her PhD research[ ], can be adapted to further improve the bound r r ∈ { , } we are able to show that T cannot be a sporadic simple group, and to rule out the case T ∼ = Alt n except in one specific action when n is a prime congruent to 3 modulo 4 (see Table 1). The proof ofTheorem 1.1 is presented in Section 7.Section 8 concludes the paper with a discussion and some open problems. In light of the growingbody of work towards a classification of point-primitive generalised quadrangles, and the possibleavenues outlined in Remark 5.11, Remark 6.4 and Section 8 for attacking the cases left open byTheorem 1.1, we feel that the following conjecture can be made with a reasonable amount of confidence. Conjecture . If a thick finite generalised quadrangle Q admits a collineation group G thatacts primitively on the point set of Q , then G is either affine or almost simple.
2. Bounding the number of points fixed by a collineation
The facts summarised in the following lemma are well known. (The existence of an order is provedin [ , Corollary 1.5.3], and proofs of assertions (i)–(iii) may be found in [ , Section 1.2].) IMPLE GROUPS, PRODUCT ACTIONS, AND GENERALISED QUADRANGLES 5
Lemma . Let Q be a thick finite generalised quadrangle. Then Q has an order ( s, t ) , and thefollowing properties hold: (i) Q has ( s + 1)( st + 1) points and ( t + 1)( st + 1) lines, (ii) s / t s t (Higman’s inequality), (iii) s + t divides st ( st + 1) . A point–line incidence geometry S = ( P , L , I ) is called a grid if there exist positive integers s and s such that P can be written as { P ij | i s , j s } , L can be written as { ℓ k | k s } ∪ { ℓ ′ k | k s } , and we have P ij I ℓ k if and only if i = k , and P ij I ℓ ′ k if andonly if j = k . Each point of S is then incident with exactly two lines, and |P| = ( s + 1)( s + 1).Let us say that such a grid has parameters s and s . Note that a grid with parameters s = s isa generalised quadrangle of order ( s , dual grid is defined analogously, by swapping the rolesof points and lines. That is, there exist positive integers t and t such that L can be written as { ℓ ij | i t , j t } , P can be written as { P i | i t } ∪ { P ′ j | j t } , P k I ℓ ij ifand only if i = k , and P ′ k I ℓ ij if and only if j = k . In this case, each line is incident with exactly twopoints, and |P| = ( t + 1) + ( t + 1). Let us say that such a dual grid has parameters t and t .If θ is a collineation of a generalised quadrangle Q = ( P , L ), then it makes sense to consider thepoint–line incidence geometry Q θ = ( P θ , L θ ) with P θ = { P ∈ P | P θ = P } , L θ = { ℓ ∈ L | ℓ θ = ℓ } , andincidence inherited from Q . Here we call Q θ the substructure of Q fixed by θ . It may happen that Q θ is a grid or a dual grid, or a generalised quadrangle. More specifically, we have the following result,based on the description of the possible structures of Q θ given by Payne and Thas [ , 2.4.1]. Lemma . Let Q = ( P , L ) be a thick finite generalised quadrangle of order ( s, t ) . Let θ be anon-identity collineation of Q , and let Q θ = ( P θ , L θ ) be the substructure of Q fixed by θ . Then at leastone of the following conditions holds. (i) P θ is empty. (ii) L θ is empty and P θ is a set of pairwise non-collinear points. In particular, |P θ | st + 1 . (iii) All points of Q θ are incident with a common line, and |P θ | s + 1 . (iv) All points of Q θ are collinear with a common point, and |P θ | s ( t + 1) + 1 . (v) Q θ is a grid. In this case, either |P θ | = ( s + 1) and s t , or |P θ | < s . (vi) Q θ is a dual grid, and |P θ | t + 1) . (vii) Q θ is a thick generalised quadrangle, and |P θ | ( s + 1)( t + 1) .In particular, either |P θ | ( s + 1)( t + 1) ; or s > t + 3 , Q θ is a grid and |P θ | < s . Proof.
The possible structures (i)–(vii) of Q θ are given by [ , 2.4.1]. We verify the claimed upperbounds for |P θ | . The bounds in cases (iii) and (iv) are immediate, because every line of Q is incidentwith exactly s + 1 points, and every point of Q is incident with exactly t + 1 lines. For case (ii), note[ , Section 2.7] that the maximum size of a set of pairwise non-collinear points in Q is st + 1. For(v), if Q θ is a dual grid with parameters t and t , then t t and t t , and hence |P θ | t + 1).Now suppose that Q θ is a grid with parameters s and s , noting that s s and s s , andassuming (without loss of generality) that s > s . If s = s = s then |P θ | = ( s + 1) , and Q θ is a generalised quadrangle of order ( s, , 2.2.2(i)] implies that s t . The case s = s − θ fixes s points incident with a line then it must also fix the final point; andif s s − |P θ | ( s + 1)( s − < s . Finally, suppose that Q is a thick finite generalisedquadrangle, and let ( s ′ , t ′ ) denote its order. Then |P θ | = ( s ′ + 1)( s ′ t ′ + 1) by Lemma 2.1(i). If t ′ = t then s ′ < s because θ = 1, so [ , 2.2.1] implies that s ′ t = s ′ t ′ s , and hence |P θ | ( s/t + 1)( s + 1) ( t /t + 1)( s + 1) = ( s + 1)( t + 1), where for the second inequality we use Lemma 2.1(ii). If t ′ < t thenthe dual statement of [ , 2.2.1] yields s ′ t ′ t , so |P θ | = ( s ′ + 1)( s ′ t ′ + 1) ( s + 1)( t + 1).The final assertion is deduced by comparing the upper bounds on |P θ | established in each case.We observe that |P θ | ( s + 1)( t + 1) except possibly in the second case of (v), where our bound is |P θ | < s . However, if s t + 2 then in this case we have |P θ | < s < ( s + 1)( t + 1). (cid:3) Remark . We mention a paper of Frohardt and Magaard [ , Section 1.3], in which resultsanalogous to Lemma 2.2 are obtained for generalised d -gons with d ∈ { , } (that is, generalisedhexagons and octagons). The known examples of such geometries admit point- and line-primitive JOHN BAMBERG, TOMASZ POPIEL, CHERYL E. PRAEGER actions of almost simple groups with socle D ( q ) or G ( q ) (for d = 6) and F ( q ) (for d = 8). Frohardtand Magaard use the aforementioned results to determine upper bounds for fixities of primitive actionsof groups G with generalised Fitting subgroup D ( q ), G ( q ) or F ( q ) (and they also treat the otherexceptional Lie type groups of Lie rank 1 or 2). By comparison, we instead apply Lemma 2.2 todetermine which groups might act primitively on the points of a generalised d -gon (with d = 4 in ourcase). (We remark that we have also investigated point-primitive generalised hexagons and octagons,although via different methods than in the present paper [
7, 18 ].)We now use Lemma 2.2 to prove Theorem 1.3, from which we deduce Theorem 1.2.
Proof of Theorem 1.3
Let ( s, t ) be the order of Q , and let Q θ = ( P θ , L θ ) be the substructure of Q fixed by θ . We must show that either |P θ | < |P| / , or ( s, t ) = (2 ,
4) and |P θ | = 15 for Q = Q − (5 , s = 2. By Lemma 2.2, we have either |P θ | < s or |P θ | ( s + 1)( t + 1). If |P θ | < s then |P θ | < |P| / since |P| = ( s +1)( st +1) > s t > s / by Lemma 2.1. If |P θ | ( s +1)( t +1)then it suffices to show that the function f ( s, t ) = (( s + 1)( st + 1)) / − ( s + 1)( t + 1) is positive for all s >
3, for all s / t s . This is readily checked when s ∈ { , } , so assume that s >
5. Thinkingof s and t as real variables, we have ∂f∂t ( s, t ) = ( s + 1)(4 s − h ( s, t )) h ( s, t ) , where h ( s, t ) = 5(( s + 1)( st + 1)) / . Since s and t are positive, this derivative is positive if and only if 4 s − h ( s, t ) >
0. Since s > t s , we have h ( s, t ) s ) / ( st ) / ) / s / . Hence, 4 s − h ( s, t ) > s / (4 s / − ) / ).The right-hand side of this inequality is positive if s > ( ) ( ) = ≈ . ∂f∂t ( s, t ) > s > s / t s . Since f ( s, t ) > f ( s, s / ) and f ( s, s / ) > s >
5, itfollows that f ( s, t ) > s >
5, for all s / t s .Now suppose that s = 2. Then t ∈ { , } by Lemma 2.1. There exist unique generalised quadran-gles of orders (2 ,
2) and (2 , W (3 ,
2) and the elliptic quadric Q − (5 , , 5.2.3 and 5.3.2]. The full collineation groups of these generalised quadrangles arePΓSp (2) and PΓU (2), respectively. One may use the package FinInG [ ] in the computer algebrasystem GAP [ ] to check that every non-identity collineation of W (3 ,
2) fixes at most 7 points. Since W (3 ,
2) has a total of 15 points and 15 / ≈ . >
7, the claimed inequality |P θ | < |P| / holds for ev-ery non-identity collineation θ in this case. On the other hand, there exist 36 non-identity collineationsof Q − (5 ,
2) that fix 15 points, but the total number of points of Q − (5 ,
2) is 27 and 27 / ≈ . < , Q − (5 ,
2) fixes at most 9 points. (cid:3)
Proof of Theorem 1.2.
Since the action of H on Ω is not semiregular, there exists h ∈ H \ { } fixing at least one point of Ω . Let f be the number of points of Ω fixed by h . Let θ = ( h , , . . . , ∈ N , and let f be the number of points of Q fixed by θ . If r = 1 then Theorem 1.3 implies that f = f < |P| / = | Ω | / . If r > f Q ri =2 | Ω i | = f < |P| / = ( Q ri =1 | Ω i | ) / ,so f ( Q ri =2 | Ω i | ) / < | Ω | / . Since ( Q ri =2 | Ω i | ) / > | Ω | ( r − / , it follows that f < | Ω | − r/ . Inparticular, 1 − r/ > f >
1, and so r (cid:3) We also use Lemma 2.2 to sharpen the 4 / /
125 = 0 .
752 in case (i) of Proposition 2.5 could be changed to 3 / ǫ for any ǫ > s in case (ii), but that this would nothave been useful for our arguments in Section 5. Proposition . Let Q = ( P , L ) be a finite generalised quadrangle of order ( s, t ) , let θ be anynon-identity collineation of Q , and let Q θ = ( P θ , L θ ) be the substructure of Q fixed by θ . Then either (i) |P θ | < |P| / , (ii) s ∈ { , } , t = s and Q θ is a generalised quadrangle of order ( s, s ) , or (iii) s > t + 3 , Q θ is a grid and |P θ | < s . IMPLE GROUPS, PRODUCT ACTIONS, AND GENERALISED QUADRANGLES 7
Proposition . Let Q = ( P , L ) be a finite generalised quadrangle of order ( s, t ) , let θ be anynon-identity collineation of Q , and let Q θ = ( P θ , L θ ) be the substructure of Q fixed by θ . Then either (i) |P θ | < |P| / , (ii) s < . × , or (iii) s > t + 3 , Q θ is a grid and |P θ | < s . Proposition . Let Q = ( P , L ) be a finite generalised quadrangle of order ( s, t ) , let θ be anynon-identity collineation of Q , and let P θ denote the set of points fixed by θ . Suppose that t = s + 2 .Then |P θ | < |P| / if s > , and |P θ | < |P| / if s > .
3. Centraliser orders in non-Abelian finite simple groups
Here we verify a series of lemmas about centraliser orders in non-Abelian finite simple groups,from which Proposition 1.6 is deduced. Specifically, we need to know which non-Abelian finite simplegroups T contain non-identity elements x with ‘large’ centralisers, in the sense that | C T ( x ) | > | T | − r/ for r equal to one of 1, 2 or 3. This question is readily and exactly answered for alternating groupsand sporadic simple groups in the following two lemmas. Note that we treat the Tits group F (2) ′ in Lemma 3.2 along with the sporadic groups. Lemma . Let T ∼ = Alt n with n > . Then (i) | C T ( x ) | < | T | / for all x ∈ T \ { } if and only if n , (ii) | C T ( x ) | < | T | / for all x ∈ T \ { } if and only if n , (iii) | C T ( x ) | < | T | / for all x ∈ T \ { } if and only if n . Proof. If n >
19 and x ∈ T is a 3-cycle, then | C T ( x ) | = ( n − > ( n !) / = | T | / . The remainingassertions are readily verified using GAP [ ]. (cid:3) Lemma . Let T be either a sporadic finite simple group or the Tits group F (2) ′ . Then (i) | C T ( x ) | < | T | / for all x ∈ T \ { } , (ii) | C T ( x ) | < | T | / for all x ∈ T \ { } if and only if T = Suz , Co , Fi , Fi , B or M , (iii) | C T ( x ) | < | T | / for all x ∈ T \ { } if and only if T ∼ = J . Proof.
This is readily verified upon checking maximal centraliser orders in the ATLAS [ ]. (cid:3) Next we consider the exceptional Lie type groups, namely those of type E , E , E ε (where ε = ± ),F , F , G , G , D or B . Note that we make no attempt to check the converse of assertion (i)(although this could be done using standard references including those cited here). Lemma . Let T be a finite simple group of exceptional Lie type. (i) If T has type E , E , E ε , F or D , then there exists x ∈ T \ { } with | C T ( x ) | > | T | / . (ii) | C T ( x ) | < | T | / for all x ∈ T \ { } if and only if T has type B . Proof. (i) For T ∼ = E ( q ), F ( q ) or D ( q ), take x ∈ T to be a unipotent element of type A in thesense of [ , Tables 22.2.1 and 22.2.4] and [ ], respectively. Then | C T ( x ) | = q | E ( q ) | , q | C ( q ) | or q ( q − | C T ( x ) | > | T | / in each case. Now supposethat T ∼ = E ( q ) or E ε ( q ), and write G := Inndiag( T ). Take x ∈ T to be a unipotent element of typeA in the sense of [ , Tables 22.2.2 and 22.2.3], respectively. Then x T = x G by [ , Corollary 17.10],so | C T ( x ) | = | C G ( x ) | / | G : T | = q | D ( q ) | / gcd(2 , q −
1) or q | A ε ( q ) | / gcd(3 , q − ε ), respectively, andagain one can check that | C T ( x ) | > | T | / in each case.(ii) If T ∼ = B ( q ) then | C T ( x ) | q < ( q ( q + 1)( q − / = | T | / for all x ∈ T \ { } [ ]. Itremains to check that | C T ( x ) | > | T | / for some x ∈ T \ { } when T has type F , G or G . Inthese respective cases, take x to be a unipotent element of type ( ˜A ) , A or ( ˜A ) in the sense of [ ,Tables 22.2.5–22.2.7], so that | C T ( x ) | = q | B ( q ) | , q | A ( q ) | or q . (cid:3) Finally, we consider the finite simple classical groups. Again, we do not check the converses ofassertions (i) or (ii), remarking only that one could do so using the monograph [ ] of Burness andGiuidici, where the conjugacy classes of elements of prime order in these groups are classified. JOHN BAMBERG, TOMASZ POPIEL, CHERYL E. PRAEGER
Lemma . Let T be a finite simple classical group. (i) If T has type A εn , D n or D n with n > , or type B n or C n with n > , then there exists x ∈ T \ { } with | C T ( x ) | > | T | / . (ii) If T has type A εn with n > , type B n or C n with n > , or type D n or D n with n > , thenthere exists x ∈ T \ { } with | C T ( x ) | > | T | / . (iii) | C T ( x ) | < | T | / for all x ∈ T \ { } if and only if T ∼ = PSL ( q ) with q = 7 . Proof.
Throughout the proof, we write q = p f with p a prime and f >
1. First suppose that T hastype A . That is, T ∼ = PSL ( q ), with q >
4. The smallest non-trivial conjugacy class of T has size q ( q − ( q −
1) or q ( q −
1) according as whether p = 2, q ≡ q ≡ T has size greater than | T | / if and only if q = 7. Equivalently, | C T ( x ) | < | T | / for all x ∈ T \ { } if and only if q = 7.Now suppose that T has type A εn with n >
2. That is, T ∼ = PSL εn +1 ( q ) (where L + := L, L − := U).Let x ∈ G := PGL εn +1 ( q ) be an element of order p with one Jordan block of size 2 and n − a = n − a = 1 and a = · · · = a p = 0 in the notation of [ , Section 3.2.3].Then x ∈ T , and x T = x G by [ , Propositions 3.2.7 and 3.3.10], so by [ , Tables B.3 and B.4] we have | C T ( x ) | = | C G ( x ) | / | G : T | = d | C G ( x ) | = d q n − | GL εn − ( q ) | , where d := gcd( n + 1 , q − ε ). Therefore,(1) | C T ( x ) | = 1 d q n ( n +1) / n − Y i =1 ( q i − ε i ) and | T | = 1 d q n ( n +1) / n +1 Y i =2 ( q i − ε i ) . For n ∈ { , } we must show that | C T ( x ) | > | T | / . If n = 2 then d
3, so | C T ( x ) | > q ( q − ε )while | T | q ( q − ε )( q − ε ). This implies that | C T ( x ) | > | T | / for all q >
7, and one may checkdirectly that this inequality also holds for q <
7. If n = 3 then d
4, so | C T ( x ) | > q ( q − ε )( q − ε )while | T | q ( q − ε )( q − ε )( q − ε ). This implies that | C T ( x ) | > | T | / for all q >
3, and a directcalculation shows that this inequality also holds for q = 2. Now suppose that 4 n
8. We mustshow that | C T ( x ) | > | T | / . Since q >
2, (1) gives | C T ( x ) | > d q n n − and | T | d (cid:16) (cid:17) n q n +2 n , and so it suffices to show that q n − n > d n − n . Indeed, since d n + 1, it suffices to show that q n − n > ( n + 1) n − n . This inequality holds for all q > n ∈ { , } , for all q > n = 6,for all q > n = 5, and for all q >
11 if n = 4. In the remaining cases, where ( n, q ) = (6 , , ,
3) or (4 , q ) with q <
11, one may check directly that | C T ( x ) | > | T | / . It remains to show that | C T ( x ) | > | T | / for all q > n >
9. If q > | C T ( x ) | > d (cid:16) (cid:17) n − q n and | T | d (cid:16) (cid:17) n q n +2 n , so it suffices to show that q n − n > d · n +5 n − . Indeed, since d n + 1, we can just show that q n − n > ( n + 1)2 n +5 n − . This inequality holds for all q > n >
11; if n = 10, it holds for all q >
5, and if n = 9, it holds for all q >
29. For n = 10 with 2 q < n = 9 with 2 q <
29, onemay check directly that | C T ( x ) | > | T | / . Finally, we must check that | C T ( x ) | > | T | / when q = 2and n >
9. Since n >
9, and since q i q i − ε q i for i >
8, (1) gives | C T ( x ) | > d (cid:16) (cid:17) n − q n −
28 7 Y i =1 ( q i −
1) and | T | d (cid:16) (cid:17) n − q n +2 n −
27 7 Y i =2 ( q i − . Noting also that d
3, we see that it suffices to show that2 n − n − n −
40 7 Y i =2 ( q i − > · n − n − . This inequality holds for all n >
9, and so the proof of the A εn case is complete.Next, suppose that T has type C n , where n >
2. That is, T ∼ = PSp n ( q ). Write G := PGSp n ( q ),noting that | G : T | = gcd(2 , q − p >
2, take x ∈ G of order p with one Jordan block of size 2 and IMPLE GROUPS, PRODUCT ACTIONS, AND GENERALISED QUADRANGLES 9 n −
1) Jordan blocks of size 1. That is, a = 2( n − a = 1 and a = · · · = a p = 0 in the notation of[ , Section 3.4.3]. Then x ∈ T , and by [ , Proposition 3.4.12], x G splits into two T -conjugacy classesand hence | C T ( x ) | = 2 | C G ( x ) | / | G : T | = q n − | Sp n − ( q ) | . If p = 2 then T = G and we take x to be an involution of type b as in [ , Table 3.4.1], so that | C T ( x ) | = q n − | Sp n − ( q ) | . Hence, forevery p , we have(2) | C T ( x ) | = 1 d q n n − Y i =1 ( q i −
1) and | T | = 1 d q n n Y i =1 ( q i − , where d = gcd(2 , q −
2. If n = 2 then | C T ( x ) | > q ( q −
1) and | T | q ( q − q − | C T ( x ) | > | T | / for all q >
2. Similarly, for n ∈ { , } one may check that | C T ( x ) | > | T | / for all q >
2. Now suppose that n >
5. Since q i >
4, we have q i − > q i for all i >
1, and so | C T ( x ) | > ( ) n − q n − n , while | T | < q n + n . Hence, to show that | C T ( x ) | > | T | / , itsuffices to show that ( ) n − q n − n >
2. This inequality holds for all q > n >
6, and for all q > n = 5; for ( n, q ) = (5 ,
2) and (5 , | C T ( x ) | > | T | / .Now suppose that T has type B n , where n >
2. That is, T ∼ = Ω n +1 ( q ) with q odd. For q ≡ x ∈ G := PGO n +1 ( q ) be an involution of type t n or t ′ n , respectively, in the sense of [ ,Sections 3.5.2.1 and 3.5.2.2]. Then x ∈ T and x T = x G , so | C T ( x ) | = | C G ( x ) | / | G : T | = | C G ( x ) | = | SO ± n ( q ) | by [ , Table B.8]. Now,(3) | SO ± n ( q ) | = q n − n ( q n ∓ n − Y i =1 ( q i − > q n n − Y i =1 ( q i − , and the right-hand side above is the value of | C T ( x ) | that we obtained in the C n case. Since | Ω n +1 ( q ) | = | PSp n ( q ) | , we therefore reach the same conclusions as for type C n .Now suppose that T has type D εn , namely T ∼ = PΩ ε n ( q ) with n >
4. Let G := Inndiag(PΩ ε n ( q )),as defined on [ , p. 56]. Assume first that p >
2, noting that | G : T | divides 4. Take x ∈ G of order p with one Jordan block of size 2( n −
2) and two Jordan blocks of size 2. That is, a = 2( n − a = 2and a = · · · = a p = 0 in the notation of [ , Section 3.5.3]. Then x ∈ T , and [ , Propositions 3.5.14(i)and (ii,b)] imply that x T = x G . Therefore, [ , Table B.12] gives | C T ( x ) | = | C G ( x ) | / | G : T | > | C G ( x ) | = q n − | O ε n − ( q ) || Sp ( q ) | , where the value of ε = ± depends on n and q as describedthere. Multiplying the inequality in (3) by 2 to get a lower bound for | O ε n − ( q ) | , it follows that(4) | C T ( x ) | > q n − ( q − n − Y i =1 ( q i − , while | T | = 1 d q n ( n − ( q n − ε ) n − Y i =1 ( q i − , where d = gcd(4 , q n − ε ). Since q i > i >
1, and in particular q n > = 81, we have(5) | C T ( x ) | > (cid:16) (cid:17) n − q n − n +6 and | T | < q n − n . For 4 n | C T ( x ) | > | T | / , so by (5) it suffices to show that ( ) n − q n − n +30 > ( ) , which holds unless ( n, q ) = (4 ,
3) or (4 , n, q ) = (4 , | C T ( x ) | > | T | / ;for ( n, q ) = (4 , GAP [ ] calculation shows that there exist elements x ∈ T \ { } for which thisinequality holds. For n > | C T ( x ) | > | T | / , and we now have q n > = 19683, so wecan replace the in (5) by to see that it suffices to show that ( ) n − q n − n +30 > ( ) .If n >
10 then this inequality holds for all q >
3, and if n = 9 then it holds for q > n = 9 and q < | C T ( x ) | > q n − n ( q n − − q − Q n − i =1 ( q i − | C T ( x ) | > | T | / except when q = 3 and ε = +. However, in this case we have | G : T | = 2(compare [ , Figure 2.5.1 and Lemma 2.2.9], noting that the discriminant of a hyperbolic quadraticform on F nq with ( n, q ) = (9 ,
3) is ⊠ , in the notation used there, because n ( q − / in the above estimate for | C T ( x ) | may be replaced by , and we again obtain | C T ( x ) | > | T | / .Finally, suppose that T ∼ = PΩ ε n ( q ) with q even, noting that T = G in this case. Take x ∈ G to be an involution of type a as in [ , Table 3.5.1]. Then | C T ( x ) | = q n − | Ω ε n − ( q ) || Sp ( q ) | and gcd(4 , q n − ε ) = 1, so instead of (4) we have(6) | C T ( x ) | > q n − ( q − n − Y i =1 ( q i −
1) and | T | = q n ( n − ( q n − ε ) n − Y i =1 ( q i − . (In the bound for | C T ( x ) | we drop a factor of because | G : T | = 1 for q even, but pick up a factorof because Ω ε n − ( q ) has index 2 in SO ε n − ( q ).) For 4 n | C T ( x ) | > | T | / . Since q i > i >
1, and in particular q n >
16, it suffices to show that ( ) n − q n − n +30 > ( ) .This inequality holds unless ( n, q ) = (4 ,
2) or (4 , | C T ( x ) | > | T | / . For n > | C T ( x ) | > | T | / . We now have q n > ) n − q n − n +30 > ( ) . This inequality holds unless ( n, q ) = (10 , n = 9 and q . One may use (6) to check that | C T ( x ) | > | T | / in each of these cases except( n, q ) = (9 , (cid:3)
4. Quasiprimitive point actions of type SD or CD
We now apply Corollary 1.5 to permutation groups N that arise as subgroups of certain typesof primitive groups. In some cases, we are also able to treat quasiprimitive groups, namely those inwhich every non-trivial normal subgroup is transitive. In this section, we consider the case where thegroup N in Corollary 1.5 has a ‘diagonal’ action. Specifically, we work under the following hypothesis. Hypothesis . Let T be a non-Abelian finite simple group, let k > , and write H = T k . Let Ω = { ( y , . . . , y k − , | y , . . . , y k − ∈ T } H , and let H act on Ω by (7) ( y , . . . , y k − , ( x ,...,x k ) = ( x − k y x , . . . , x − k y k − x k − , . Suppose that N = H r is a collineation group of a thick finite generalised quadrangle Q = ( P , L , I ) oforder ( s, t ) , such that P = Ω r and N has the product action on P . This situation arises when N is the socle of a primitive permutation group G Sym(Ω) of typeHS, HC, SD or CD. For type HS (respectively HC) we have k = 2 and r = 1 (respectively r > G has two minimal normal subgroups, each isomorphic to T r , and the socle of G is T r × T r , whichis isomorphic to N . For type SD (respectively CD) we have k > r = 1 (respectively r > G has a unique minimal normal subgroup, isomorphic to T kr ∼ = N . Note that the notation k and r is consistent with that of Table 1. Of course, G must (usually) satisfy certain other conditions[ , Section 6] in order to actually be primitive, but these conditions are not needed for the proof ofProposition 4.2. It suffices that there is a subgroup of the form N . In particular, we are also ableto treat quasiprimitive groups [ , Section 12], because the (action of the) socle of G is the same asin the respective primitive types. (Note that a quasiprimitive group of type HS or HC is necessarilyprimitive, but a quasiprimitive group of type SD or CD need not be primitive.)Proposition 4.2 shows, in particular, that the parameter r in Hypothesis 4.1 can be at most 3.As illustrated after Corollary 1.5, the proof relies on the information about centraliser orders in non-Abelian finite simple groups given in Proposition 1.6. We also observe that when r = 3, there alwaysexists a solution ( s, t ) = ( | Ω | − , | Ω | + 1) of the equation | Ω | = | Ω | r = |P| = ( s + 1)( st + 1), and thissolution satisfies properties (ii) and (iii) of Lemma 2.1. Hence, although we are unable to rule out thecase r = 3 completely, we verify that this ‘obvious’ situation cannot occur. Proposition . If Hypothesis 4.1 holds then r and | C T ( x ) | < | T | − r/ for all x ∈ T \ { } ,and in particular T must appear in Table . Moreover, if r = 3 then ( s, t ) = ( | Ω | − , | Ω | + 1) . Proof.
Note first that |P| = | Ω | r = | T | ( k − r . In particular, the excluded case ( s, t ) = (2 ,
4) inCorollary 1.5 does not arise, because |P| > | T | > | Alt | = 60 > (2 + 1)(2 · x := x = · · · = x k = 1 in (7), then ( y , . . . , y k − , ∈ Ω is fixed if and only if y , . . . , y k − ∈ C T ( x ). That is,( x, . . . , x ) ∈ H fixes precisely | C T ( x ) | k − elements of Ω (and, in particular, the action of H on Ω is notsemiregular). Corollary 1.5 therefore implies that r | C T ( x ) | k − < | Ω | − r/ = | T | ( k − − r/ ,namely | C T ( x ) | < | T | − r/ , for all x ∈ T \ { } . If r = 4 then we have a contradiction becauseevery non-Abelian finite simple group T contains a non-identity element x with | C T ( x ) | > | T | / . For IMPLE GROUPS, PRODUCT ACTIONS, AND GENERALISED QUADRANGLES 11 example, it is well known that every non-Abelian finite simple group T contains an involution x with | C T ( x ) | > | T | / (in fact, every involution has this property [ , Proposition 2.4]). Therefore, r T must be one of the groups appearing in Table 2. Toprove the final assertion, suppose towards a contradiction that r = 3 and ( s, t ) = ( | Ω | − , | Ω | + 1).Take any x ∈ T with | C T ( x ) | > | T | / . Then (( x, . . . , x ) , (1 , . . . , , (1 , . . . , ∈ H r = H = N fixes | C T ( x ) | k − | T | k − > | T | k − / = |P| / points of Q , contradicting Proposition 2.6. (cid:3) The following immediate consequence of Proposition 4.2 (and the preceding observations) impliesthe SD and CD cases of Theorem 1.1.
Proposition . Let Q = ( P , L ) be a thick finite generalised quadrangle admitting a collineationgroup G that acts quasiprimitively of type SD or CD on P . Then the conditions in Table 1 hold.
5. Primitive point actions of type HS or HC
We now consider the case where k = 2 in Hypothesis 4.1 in more detail. As explained above, thiscase arises when N is the socle of a primitive permutation group G Sym(Ω) of type HS ( r = 1) orHC ( r > k = 2 it is natural to simplify the notation of Hypothesis 4.1 by identifying the setΩ with T r , so we first re-cast the hypothesis in this way and also establish some further notation. Hypothesis . Let T be a non-Abelian finite simple group and let N = T r × T r act on T r by (8) y ( u ,u ) = u − yu . Let M = { ( u, | u ∈ T r } N , so that M may be identified with T r acting regularly on itself byright multiplication. Suppose that N is a collineation group of a thick finite generalised quadrangle Q = ( P , L , I ) of order ( s, t ) with P = T r . Let P ⊂ P denote the set of points collinear with butnot equal to the identity element ∈ T r = P , and let L ⊂ L denote the set of lines incident with .Given a line ℓ ∈ L , let ¯ ℓ ⊂ P denote the set of points incident with ℓ . The following lemma may essentially be deduced from [ , Lemma 10] upon observing that theassumption gcd( s, t ) > , p. 654] as follows: the point-regular group G is our M ∼ = T r , and the point O is our point 1, so that ∆ is our P \ { } . Lemma . Suppose that Hypothesis 5.1 holds. Let x ∈ P \ { } , and let ℓ x be the unique line in L incident with x . Then, for every i ∈ { , . . . , | x | − } , the conjugacy class ( x i ) T r is contained in P .Moreover, the collineation ( x, ∈ M fixes ℓ x . Proof.
Let us first establish some notation. Given u ∈ T r = P , writefix P ( u ) = { P ∈ P | P ( u, = P } , coll P ( u ) = { P ∈ P | P ( u, is collinear with but not equal to P } , fix L ( u ) = { ℓ ∈ L | ℓ ( u, = ℓ } , conc L ( u ) = { ℓ ∈ L | ℓ ( u, is concurrent with but not equal to ℓ } . Since the subgroup M = { ( u, | u ∈ T r } of N acts regularly on P , fix P ( u ) is empty. Moreover, P ∈ coll P ( u ) if and only if P ( P − , = 1 and ( P u ) ( P − , = P uP − are collinear, which is if and onlyif P uP − ∈ u T r ∩ P . Since for g, h ∈ T r we have gug − = huh − if and only if g − h ∈ C T r ( u ), itfollows that | coll P ( u ) | = | u T r ∩ P || C T r ( u ) | , as in the proof of [ , Lemma 3]. Then (again, as in that proof) [ , 1.9.2] implies that(9) | coll P ( u ) | = ( s + 1) | fix L ( u ) | + | conc L ( u ) | = | u T r ∩ P || C T r ( u ) | (for every u ∈ T r ) . Now, since x ∈ P , we have u − xu = x ( u,u ) ∈ P for every collineation of the form ( u, u ) ∈ N ,because such collineations (are precisely those that) fix the point 1. That is, every T r -conjugate of x is in P . In other words, x T r ∩ P = x T r , and so (9) implies that(10) | coll P ( x ) | = ( s + 1) | fix L ( x ) | + | conc L ( x ) | = | x T r || C T r ( x ) | = | T r | = |P| = ( s + 1)( st + 1) . In particular, we have coll P ( x ) = P ; that is, every point of Q is mapped to a collinear point under thecollineation ( x, ∈ M . We now claim that conc L ( x ) is empty. If not, then some line ℓ is concurrentwith its image under the collineation ( x, P denote the unique point incident with both ℓ and ℓ ( x, . Then P x − is incident with ℓ , being the image of P under the collineation ( x, − = ( x − , P x − = P because x = 1 and M acts regularly on P . Since Q is thick, there exists a thirdpoint P incident with ℓ , distinct from P and P x − . Since coll P ( x ) = P , the points P ( x, = P x and P are collinear. Moreover, P x is collinear with P , because both of these points are incident with ℓ ( x, . Therefore, P x is collinear with two distinct points that are incident with ℓ , namely P and P , and so P x is itself incident with ℓ because Q contains no triangles. This, however, means that P x is incident with both ℓ and ℓ ( x, , which forces P x = P and hence P = P x − , a contradiction.Therefore, | conc L ( x ) | = 0 as claimed, and so (10) implies that(11) | fix L ( x ) | = st + 1 . Next, we show that ( x i ) T r ⊆ P for all i ∈ { , . . . , | x | − } . For each such i , we certainly havefix L ( x ) ⊆ fix L ( x i ), because if the collineation ( x,
1) fixes a line then so too does ( x, i = ( x i , | fix L ( x i ) | > | fix L ( x ) | = st + 1, by (11). On the other hand, no two lines fixed by ( x i ,
1) canbe concurrent, because if they were, then the unique point incident with both lines would be fixed by( x i , M acts regularly on P . Hence, the total number of points that are incidentwith some line in fix L ( x i ) is ( s + 1) | fix L ( x i ) | . As this number cannot exceed |P| = ( s + 1)( st + 1), wemust also have | fix L ( x i ) | st + 1. Therefore, | fix L ( x i ) | = st + 1. Now (9) implies, on the one hand,that | coll P ( x i ) | = ( s + 1) | fix L ( x i ) | + | conc L ( x i ) | = |P| + | conc L ( x i ) | . Since | coll P ( x i ) | |P| , this implies that | conc L ( x i ) | = 0, and then in turn that |P| = | coll P ( x i ) | .Appealing again to (9), we now deduce that | ( x i ) T r ∩ P || C T r ( x i ) | = |P| = | T r | , which implies that( x i ) T r ⊆ P as required. The first assertion is therefore proved.Finally, we must show that the collineation ( x,
1) fixes the unique line ℓ x ∈ L incident with x . If | x | = 2, then ( x,
1) fixes ℓ x because it fixes setwise the subset { , x } of points incident with ℓ x . Thatis, it maps 1 to 1 ( x, = 1 x = x and x to x ( x, = x = 1. Now suppose that | x | >
2. Then the point1 ( x, = x = 1 is collinear with x because x is collinear with 1. On the other hand, ( x ) T r ⊆ P by thefirst assertion, so in particular x is collinear with 1. Therefore, x is collinear with two distinct pointsincident with ℓ x (namely 1 and x ), and so is itself incident with ℓ x because Q contains no triangles.Hence, ( x,
1) fixes ℓ x because it maps two points incident with ℓ x , namely 1 and x , to another twopoints incident with ℓ x , namely x and x . (cid:3) Hypothesis 5.1 imposes the following restrictions on the points and lines incident with the identityelement of T r = P , and on the order ( s, t ) of Q . Lemma . The following assertions hold under Hypothesis 5.1. (i) P is a union of T r -conjugacy classes. (ii) Every line ℓ ∈ L has the property that ¯ ℓ is a subgroup of T r . Specifically, ¯ ℓ = { u ∈ T r | ( u, ∈ M fixes ℓ } . (iii) Every line ℓ ∈ L is incident with an involution. (iv) If some line in L is incident with representatives of every T r -conjugacy class of involutionsin P , then N acts transitively on the flags of Q and r > . (v) T r has at least three conjugacy classes of involutions. (vi) If T r has exactly three conjugacy classes of involutions, then either P contains exactly twoof these classes, or N acts transitively on the flags of Q and r > . (vii) gcd( s, t ) = 1 and t > s + 1 . Proof. (i) This follows immediately from Lemma 5.2.(ii) If u ∈ ¯ ℓ then the collineation ( u, ∈ M fixes ℓ by Lemma 5.2. Conversely, if ( u,
1) fixes ℓ then, because 1 ∈ P is incident with ℓ , so too is 1 ( u, = u ; that is, u ∈ ¯ ℓ . IMPLE GROUPS, PRODUCT ACTIONS, AND GENERALISED QUADRANGLES 13 (iii) If ℓ ∈ L is not incident with any involution, then ¯ ℓ , which is a subgroup of T r by (ii), musthave odd order. That is, s + 1 = | ¯ ℓ | must be odd. However, ( s + 1)( st + 1) = | T | r is even by theFeit–Thompson Theorem [ ], so s must be odd and hence s + 1 must be even, a contradiction.(iv) If ℓ ∈ L is incident with representatives of every conjugacy class of involutions in P , then ℓ can be mapped to any other line in L by some element of the stabiliser N = { ( u, u ) | u ∈ T r } in N of the point 1 ∈ P = T r . Since N acts transitively on P , this means that N acts transitively on theflags of Q . If r = 1, this contradicts the main result of our earlier paper [ , Theorem 1.1], so r > T r contains at most two conjugacy classes of involutions.Then r = 1, because if r > x ∈ T gives rise to the three pairwise non-conjugateinvolutions ( x, , . . . , , x, , . . . ,
1) and ( x, x, , . . . ,
1) in T r . Hence, by (iii), T must have exactlytwo conjugacy classes of involutions, say x T and y T , and both must be contained in P . Without lossof generality, x and y commute, because at least one of them centralises a Sylow 2-subgroup of T .Therefore, xy is an involution, and so must be collinear with 1 ∈ P . Since 1 is collinear with x , theimages of 1 and x under the collineation ( y, ∈ M are collinear. That is, 1 ( y, = y is collinear with x ( y, = xy . Similarly, 1 and y are collinear, and hence so too are 1 ( x, = x and y ( x, = yx = xy .Since the involution xy is also collinear with 1 and Q contains no triangles, the points 1, x , y and xy must be incident with a common line. In particular, x and y are incident with a common line in L .Since r = 1, this contradicts (iv).(vi) Let x , y and z denote representatives of the three T r -conjugacy classes of involutions. If P contains exactly one of these classes, then N acts flag-transitively by (iv), and it follows from [ ,Theorem 1.1] that r >
2. Now suppose that P contains all three of x T , y T and z T . Without loss ofgenerality, x centralises a Sylow 2-subgroup of T r and both y and z commute with x , so xy = yx and xz = zx are involutions. Arguing as in the proof of (iii), we deduce that 1, x , y and xy are incidentwith a common line ℓ ∈ L . Replacing y by z in this argument, we see that z is also incident with ℓ ,so (iv) again implies that N acts flag-transitively (and it follows as above that r > s, t ) > , Lemma 6(i)] implies that every T r -conjugacy class intersects P .However, assertion (i) then implies that P = P , which is impossible. Therefore, gcd( s, t ) = 1. Inparticular, to show that t > s + 1 it suffices to show that t > s . The proof of this assertion is adaptedfrom that of [ , Corollary 2.3]. Choose two distinct lines ℓ , ℓ ∈ L , so that ¯ ℓ and ¯ ℓ are subgroupsof T r by (ii). For brevity, we now abuse notation slightly and identify ℓ and ℓ with ¯ ℓ and ¯ ℓ ,respectively, dropping the ‘bar’ notation. Since ℓ is a subgroup of T r and right multiplication by anyelement of T r is a collineation of Q (identified with an element of M ), we have in particular that everyright coset ℓ g of ℓ with g ∈ ℓ corresponds precisely to the set of points incident with some lineof Q . Similarly, left multiplications (identified with elements of { } × T r N ) are collineations, soevery left coset g ℓ of ℓ with g ∈ ℓ is a line of Q . Therefore, L ′ = { g ℓ | g ∈ ℓ } ∪ { ℓ g | g ∈ ℓ } is a subset of L . Consider also the subset P ′ = ℓ ℓ of P = T r , and let I ′ be the restriction of Ito ( P ′ × L ′ ) ∪ ( L ′ × P ′ ). If we can show that Q ′ = ( P ′ , L ′ , I ′ ) is a generalised quadrangle of Q oforder ( s, , 2.2.2(i)] will imply that t > s . Let us first check that Q ′ satisfies the generalisedquadrangle axiom. Let ℓ ∈ L ′ and take P ∈ P ′ not incident with ℓ . Then, since Q satisfies thegeneralised quadrangle axiom, there is a unique point P ∈ P incident with ℓ and collinear with P .Since ℓ ⊂ P ′ , we have P ∈ P ′ , and so Q ′ also satisfies the generalised quadrangle axiom. It remainsto check that Q ′ has order ( s, L ′ is incident with s + 1 points in P ′ , being a coset ofeither ℓ or ℓ , so it remains to show that every point in P ′ is incident with exactly two lines in L ′ .Given P = g g ∈ P ′ , where g ∈ ℓ , g ∈ ℓ , each line ℓ ∈ L ′ incident with P is either of the form h ℓ for some h ∈ ℓ or ℓ h for some h ∈ ℓ , and since P ∈ ℓ , we must have h = g or h = g ,respectively. Therefore, P is incident with exactly two lines in L ′ , namely g ℓ and ℓ g . (cid:3) Proposition 4.2 restricts the possibilities for the simple group T in Hypothesis 5.1 to those listedin Table 2. The following result shows that, furthermore, T must be a Lie type group. Proposition . If Hypothesis 5.1 holds then T is a Lie type group. Proof.
We have |P| = | T | r , and r ∈ { , , } by Proposition 4.2. For each of the alternating andsporadic simple groups T in Table 2, we check computationally for solutions of | T | r = ( s + 1)( st + 1)satisfying s > t > r = 3 then the r T s t s ( t + 1)1 Alt
11 19 2201 Alt
19 53 10262 Alt
19 341 64982 M
89 7831 6970482 J
419 175141 73384498
Table 3.
Alternating and sporadic groups in the proof of Proposition 5.4. only such solutions have the form ( s, t ) = ( | T | − , | T | + 1) = ( | Ω | − , | Ω | + 1), and this contradictsthe final assertion of Proposition 4.2. If r ∈ { , } then the possibilities for T and ( s, t ) are as inTable 3. By Lemma 5.3(i), P is a union of T r -conjugacy classes, and so we must be able to partition |P | = s ( t + 1) into a subset of the sizes of these classes (respecting multiplicities). When r = 1 and T ∼ = Alt or Alt , this is impossible: the non-trivial conjugacy class sizes not exceeding s ( t + 1) are 70,105 and 210 in the first case, and 105, 112 and 210 in the second (with each occurring exactly once).Similarly, if r = 2 and T ∼ = J , one may check computationally that there is no partition of s ( t + 1),where ( s, t ) = (419 , T -conjugacy class sizes. Hence, it remains to considerthe cases where r = 2 and T ∼ = Alt or M . Here we first determine computationally the possiblepartitions of s ( t + 1) into non-trivial T -conjugacy class sizes to obtain a list of possible partitions P into T -conjugacy classes. Now, because the point graph of Q is a strongly regular graph in whichadjacent vertices have λ := s − µ := t + 1common neighbours, P must be a partial difference set of T with these parameters. That is, eachnon-identity element y ∈ T must have exactly λ representations of the form y = z i z − j for z i , z j ∈ P if y ∈ P , and exactly µ such representations if y
6∈ P . A computation verifies that this condition isviolated for each of the partitions of P determined in the previous step. (cid:3) Remark . In the proof of Proposition 5.4, and at several other points in Sections 5 and 6,we need to check computationally whether certain positive integers X can be equal to the numberof points of a thick finite generalised quadrangle. That is, we check for integral solutions ( s, t ) ofthe equation ( s + 1)( st + 1) = X subject to the constraints s > t > s / t s t and s + t | st ( st + 1) imposed by Lemma 2.1. In Section 5, X has the form | T | m for some non-Abelianfinite simple group T and some m
3, and in Section 6 we instead have X = Y m with m Y the index of a maximal subgroup of an almost simple group. The above inequalities imply that s mustlie between X / − X / , so it suffices to consider every integer s in this range and determinewhether t = (( X − /s − / ( s + 1) is an integer and, if so, whether s + t | st ( st + 1). We remarkthat we found it useful to also observe that s must divide X −
1, because it turned out that X − r cannot equal 3, and deduce some further restrictions on T when r ∈ { , } . Proposition . If Hypothesis 5.1 holds then r and T is a Lie type group with the propertythat | C T ( x ) | < | T | − r/ for all x ∈ T \ { } . Proof.
By Propositions 4.2 and 5.4, T is a Lie type group and r
3. We now show that | C T ( x ) | < | T | − r/ for all x ∈ T \ { } and deduce from this that r = 3. Suppose, towards a contradiction, thatthere exists x ∈ T \ { } with | C T ( x ) | > | T | − r/ . Define w = ( x, , . . . , ∈ T r and let Q θ = ( P θ , L θ )be the substructure of Q fixed by θ = ( w, w ) ∈ N . Then P θ = C T ( x ) × T r − , and hence(12) |P θ | > | T | (1 − r/ r − = | T | r/ = |P| / . Proposition 2.4 then says that either s > t + 3, or ( s, t ) ∈ { (2 , , (3 , } . The first of these conditionscontradicts Lemma 5.3(vii); the second implies that | T | r = ( s + 1)( st + 1) ∈ { , } , which isimpossible because | T | >
60. Hence, every x ∈ T \ { } must satisfy | C T ( x ) | < | T | − r/ . For r = 3this says that | C T ( x ) | < | T | / for all x ∈ T \ { } , a contradiction because we can always find some x with | C T ( x ) | > | T | / (as noted in the proof of Proposition 4.2). Therefore, r (cid:3) IMPLE GROUPS, PRODUCT ACTIONS, AND GENERALISED QUADRANGLES 15
Proposition 5.6 allows us to further reduce the list of candidates for the simple group T in Hy-pothesis 5.1 in the remaining cases r ∈ { , } . Let us first consider r = 2. Corollary . If Hypothesis 5.1 holds with r = 2 then T is of Lie type A , A ε , B or G . Inparticular, T has a unique conjugacy class of involutions. Proof.
The result is verified by straightforward calculations involving the bound on centraliser ordersimposed by Proposition 5.6, but we include the details in Appendix A. (cid:3)
We can now prove the HC case of Theorem 1.1.
Theorem . If Q is a thick finite generalised quadrangle with a collineation group G that actsprimitively on the point set P of Q , then the action of G on P does not have O’Nan–Scott type HC. Proof.
As explained above, the socle of G is a group N = T r × T r as in Hypothesis 5.1, for some r > r = 2 and that T has a unique conjugacy class of involutions.In particular, T r = T has exactly three conjugacy classes of involutions, with representatives ( x, , y ) and ( x, y ), where x and y are involutions in T . Now, [ , Theorem 1.1] says that G cannot acttransitively on the flags of Q , so in particular N cannot act transitively on the flags of Q . Lemma 5.3(vi)therefore implies that P = T must contain exactly two T -conjugacy classes. Hence, without lossof generality, P contains the class ( x, T . Since G acts primitively on P , it induces a subgroup ofAut( T ) = Aut( T ) ≀ Sym that swaps the two simple direct factors of T . Therefore, P also containsthe class (1 , y ) T , and so does not contain the class ( x, y ) T . In particular, no line ℓ ∈ L can beincident with both a conjugate of ( x,
1) and a conjugate of (1 , y ), because by Lemma 5.3(ii), ℓ wouldthen also be incident with the product of these elements, a conjugate of ( x, y ). Hence, L is partitionedinto two sets of lines: those incident with conjugates of ( x, , y ). Since G swaps these sets, G acts flag-transitively, in contradiction with [ , Theorem 1.1]. (cid:3) For r = 1 we are left with the following list of candidates for T . Corollary . Suppose that Hypothesis 5.1 holds with r = 1 . Then T is of Lie type A , A εn with n , B , C , C , D εn with n , or exceptional Lie type other than E . Proof.
By Propositions 4.2 and 5.4, T is one of the Lie type groups in the first column of Table 2.By arguing as in the proof of Proposition 5.6 but applying Proposition 2.5 instead of Proposition 2.4in the first paragraph, we conclude that one of the following conditions must also hold:(i) every non-identity element x ∈ T satisfies | C T ( x ) | < | T | / , or(ii) s . × .By choosing appropriate elements x ∈ T as in the proofs in Section 3, we are able to use this to deducethat T does not have type A ε , A ε , B , C , D ε , D ε or E . We rule out E here as an example, and includedetails of the remaining cases in Appendix A. If T ∼ = E ( q ) then ( s + 1) > |P| = | T | > | E (2) | ≈ . × and hence s > | E (2) | / − ≈ . × , contradicting (ii), so (i) must hold. However,as noted in the proof of Lemma 3.3, there exists x ∈ T ∼ = E ( q ) with | C T ( x ) | = q | E ( q ) | ∼ q ,while | T | / ∼ ( q ) / < q . Indeed, one may check that | C T ( x ) | > | T | / for all q > (cid:3) Finally, we use Lemma 5.3 to reduce the list of candidates for T in Corollary 5.9 to those given inthe first row of Table 1, thereby proving the HS case of Theorem 1.1. Proposition . Let Q = ( P , L ) be a thick finite generalised quadrangle admitting a collineationgroup G that acts primitively of type HS on P , with socle T × T for some non-Abelian finite simplegroup T . Then T has Lie type A ε , A ε , B , C , C , D ε , D ε , D ε , E ε , E or F . Proof.
We are assuming that Hypothesis 5.1 holds with r = 1, so T must be one of the groups listedin Corollary 5.9. It remains to show that, further, T cannot have Lie type A , A ε , A ε , A ε , B , G , F , G or D . This follows from Lemma 5.3(v), because in each of these cases T has at most twoconjugacy classes of involutions. (This may be verified using, for example, [ , Table 4.5.1] for oddcharacteristic and [ ] for even characteristic.) (cid:3) Remark . Proposition 5.10 begs the obvious question of whether we can rule out the lastremaining candidates for T listed there. We are confident that we will eventually be able to do this, butit seems that it will require even more new ideas and a detailed case-by-case analysis. Of course, someof the remaining groups can be ruled out in certain cases using Lemma 5.3(v); in particular, if T hasLie type C , C , F or E ε in characteristic p , then we must have p = 2, because in odd characteristicthese groups have only two conjugacy classes of involutions. When T has exactly three conjugacyclasses of involutions, we can begin by applying Lemma 5.3(v) (because we know from [ ] that N cannot act transitively on the flags of Q ), and then the arguments in the proof of Lemma 5.3(iv–vi) can be extended to deduce some restrictions on which involutions can appear in P . However,even with this extra information, we have thus far been unable to completely rule out any of theremaining candidates for T . These kinds of arguments become more difficult when T has more thanthree conjugacy classes of involutions, and in any case, it seems that it will be necessary to treat eachgroup individually, and to use the structure of its involution centralisers in some detail. Although notan ideal state of affairs, we therefore leave the remaining cases for a future project.
6. Primitive point actions of type PA
We now apply Corollary 1.5 to the case where N is the socle of a primitive permutation group G of O’Nan–Scott type PA. The notation of Hypothesis 6.1 coincides with that of Table 1. Hypothesis . Let Q = ( P , L ) be a thick finite generalised quadrangle of order ( s, t ) admittinga collineation group G that acts primitively of type PA on P , writing T r G H ≀ Sym r for somealmost simple primitive group H Sym(Ω) with socle T , where r > . Proposition . If Hypothesis 6.1 holds then r and every non-identity element of H fixes less than | Ω | − r/ points of Ω . Proof.
The socle of G is N = T r and the action of H on Ω is not semiregular, so the result followsimmediately from Corollary 1.5. (cid:3) To say more than this, we would like to have generic lower bounds for the fixity f ( H ), namelythe maximum number of fixed points of a non-identity element, of an almost simple primitive group H Sym(Ω). This problem was investigated in a recent paper of Liebeck and Shalev [ ], who provedthat f ( H ) > | Ω | / except in a short list of exceptions. This lower bound is not quite large enoughto force further restrictions on r in Proposition 6.2, because to rule out r = 4 (as we did for typesHC and CD) we would need f ( H ) to be at least | Ω | / . However, Liebeck and Shalev remark (after[ , Theorem 4]) that their | Ω | / bound could potentially be improved generically to around | Ω | / ,which would be sufficient for this purpose. Work in this direction is currently being undertaken byElisa Covato at the University of Bristol as part of her PhD research [ ], with the aim of classifyingthe almost simple primitive permutation groups H Sym(Ω) containing an involution that fixes atleast | Ω | / points. As of this writing, the alternating and sporadic cases have been completed, andso we are able to apply these results to sharpen Proposition 6.2 as follows. Proposition . Suppose that Hypothesis 6.1 holds with r > and T an alternating group ora sporadic simple group, and let S H denote the point stabiliser in the action of H on Ω . Then r ∈ { , } , H = T ∼ = Alt p with p a prime congruent to modulo , and S ∩ T = p. p − . Proof.
Since r >
2, Proposition 6.2 tells us that r ∈ { , } , and that the fixity f ( H ) of H must be atmost | Ω | − r/ . If f ( H ) | Ω | − / = | Ω | / then Covato’s results [ ] imply that either (i) T ∼ = Alt p with p ≡ S ∩ T = p. p − , or (ii) H and S are in Table 4.In case (i) we can at least deal with the situation where H = Sym p . Indeed, by the argument in[ , Section 6], there is an involution u ∈ S = p. ( p −
1) fixing 2 ( p − / ( p − )! elements of Ω, which isgreater than | Ω | / = (2( p − / provided that p >
7. If p = 7 then we observe that u still fixesmore than | Ω | / elements. This rules out r = 4, because then 1 / > − r/ /
5. For r = 3we apply Proposition 2.6. We have | Ω | = 2 ·
5! = 120 and hence |P| = | Ω | = 120 , and the onlysolution of 120 = ( s + 1)( st + 1) satisfying s > t > s, t ) = (119 , Q fixes at most IMPLE GROUPS, PRODUCT ACTIONS, AND GENERALISED QUADRANGLES 17
H S H S
Alt : SL (3) J . . . . . . . : 2 . PSL (7) Th 31 . . : 2 . PSL (7) . . . PSL (2) Table 4.
Actions with small fixity in Proposition 6.3. |P| / points. However, the collineation ( u, , ∈ G fixes more than | Ω | / | Ω | = |P| / points, acontradiction. Therefore, if we are in case (i) then we must have H = Alt p , as per the assertion.Now suppose that H and S are in Table 4. First consider the six cases on the left-hand side ofthe table. In each of these cases, f ( H ) is at least | Ω | / , so r = 4 is ruled out. For r = 3, we applyProposition 2.6 as above. Since f ( H ) > | Ω | / , we have in particular f ( H ) > | Ω | / . Choose u ∈ H fixing at least | Ω | / elements of Ω, and consider the collineation ( u, , ∈ G , which fixes at least | Ω | / = | Ω | / = |P| / points of Q . Since the only solutions of | Ω | = ( s + 1)( st + 1) satisfying s > t > t = s +2, Proposition 2.6 providesa contradiction. Now consider the five cases on the right-hand side of Table 4. The actions of J . . | Ω | / [ , Lemma 5.3], so these are ruled out for both r = 3and r = 4 by the same arguments as above. Now consider the action of M . Here | Ω | = 40320, and for r = 4 there are no solutions of | Ω | r = ( s + 1)( st + 1) satisfying s > t > r = 3, the only solution is ( s, t ) = (40319 , , Lemma 5.3], we have f ( H ) = 5, realised by an element u of order 11, and so we can construct a collineation θ = ( u, , ∈ G of Q fixing 5 | Ω | = 8128512000 points. However, s = t − < t + 3, so the final assertion of Lemma 2.2implies that |P θ | ( s + 1)( s + 3) = 1988752683 < | Ω | = 3843461129719173164826624000000. For r = 4there is no admissible solution of | Ω | r = ( s + 1)( st + 1). For r = 3 the only admissible solutionis ( s, t ) = ( | Ω | − , | Ω | + 1), and so the final assertion of Lemma 2.2 implies that any non-identitycollineation of Q fixes at most ( | Ω | + 1)( | Ω | + 3) points. However, [ , Lemma 5.3] tells us that f ( H ) = 22, so we can construct a collineation with 22 | Ω | points to yield a contradiction. (cid:3) Remark . Further improvements to Proposition 6.2 will be made in a future project. In thefirst instance, we hope to use Covato’s results [ ] on fixities of Lie type groups (once available), tocomplete our treatment of the cases r = 3 and r = 4. We also note that it is straightforward to handlethe case r = 2 with T a sporadic simple group, and likewise the almost simple case with sporadicsocle, computationally along the lines of [ , Section 6] (but assuming only point-primitivity and notline-primitivity). However, we omit these computations from the present paper for brevity.
7. Proof of Theorem 1.1
Let us now summarise the proof of Theorem 1.1. If the primitive action of G on P has O’Nan–Scott type AS or TW, then the conditions stated in Table 1 follow immediately from Theorem 1.3.Types HS, HC and PA are treated in Proposition 5.10, Theorem 5.8 and Proposition 6.3, respectively.Types SD and CD are treated together in Proposition 4.3.
8. Discussion and open problems
We feel that the results presented in this paper represent a substantial amount of progress towardsthe classification of point-primitive generalised quadrangles, but there is evidently still a good dealof work to do. We conclude the paper with a brief discussion, and outline some open problems thatcould be investigated independently and then potentially applied to our classification program.As discussed in Remark 5.11, we are confident that we will eventually be able to finish the HS case,and it is at least somewhat clear how this might be done. The SD and CD cases would appear to be more difficult, however. The arguments used in Section 5 do not work in these cases, because the proofof Lemma 5.2 (and therefore Lemma 5.3) relies in a crucial way on having k = 2 in Hypothesis 4.1,so that conjugation by an element of the underlying point-regular group M is a collineation. We havethus far been unable to find a way to work around this difficulty in any sort of generality. On the otherhand, a primitive (respectively quasiprimitive) group of type SD must induce a primitive (respectivelytransitive) permutation group on the set of simple direct factors of its socle T k , and it seems that itshould be possible to use this extra structure to say more about the SD and CD types, at least inthe primitive case (especially since we have already reduced the list of candidates for T to those inTable 1). Although we have made some preliminary investigations along these lines, we do not yetknow how to finish the SD and CD cases, and so we leave this task for a future project. There is, ofcourse, a potential — but apparently extremely challenging — way to deal with all of the types HS, SDand CD, and also with type TW, in one fell swoop. In each of these cases, the full collineation groupmust have a point-regular subgroup of the form T m , for some m , with T a non-Abelian finite simplegroup. Hence, it would certainly be sufficient to show that such a group cannot act regularly on thepoints of a generalised quadrangle. However, this would appear to be a very difficult problem in light ofthe (limited) existing literature. Yoshiara [ ] managed to show that a generalised quadrangle of order( s, t ) with s = t cannot admit a point-regular group, while Ghinelli [ ] considered the case where s is even and t = s , showing that such a group must have trivial centre and cannot be a Frobeniusgroup. Beyond this, not much else seems to be known in the way of restrictions on groups that can actregularly on the points of a generalised quadrangle (though certainly many of the known generalisedquadrangles admit point-regular groups [ ], and the Abelian case is understood [ ]). AlthoughYoshiara [ ] has an extensive suite of lemmas that one might attempt to use to investigate (inparticular) the possibility that a group of the form T n acts point-regularly on a generalised quadrangle Q , the bulk of these lemmas assume that the order ( s, t ) of Q satisfies gcd( s, t ) = 1. Although thiscondition holds under Yoshiara’s intended assumption that s = t , it seems to be difficult to guaranteein general. Indeed, according to Lemma 5.3(ii) (and perhaps not surprisingly), it must fail in our HScase. On the other hand, one might seek a contradiction by examining the arithmetic nature of theequation | T | m = ( s + 1)( st + 1) subject to the constraints s > t > s / t s t and s + t | st ( st + 1) imposed by Lemma 2.1, and asking when it can be guaranteed that a solution mustsatisfy gcd( s, t ) = 1. More generally, one might simply ask whether this equation can have any suchsolutions at all. This motivates the following problem. Problem . Determine for which non-Abelian finite simple groups T , and which positive integers m , there exist integral solutions ( s, t ) of the equation (13) | T | m = ( s + 1)( st + 1) with s > , t > , s / t s t and s + t | st ( st + 1) . Failing this, determine when such a solution must satisfy gcd( s, t ) = 1 . As noted before Proposition 4.2, there is always an ‘obvious’ solution of (13) when m is divisibleby 3, namely ( s, t ) = ( | T | m/ − , | T | m/ + 1), and gcd( s, t ) = 1 in this case because | T | is even. Itwould be useful even to know whether this is the unique solution in this particular situation. We doknow that (13) has solutions for certain T when m = 1 or 2, as demonstrated by Table 3, but wedo not recall encountering any solutions apart from the aforementioned ‘obvious’ ones when m > T and values of m will never yield a solution of (13). For example, if T ∼ = PSL ( q ) and m = 1 then there is no solution if q < , but we do not see how to go about proving that there isno solution for any q .One might also ask about gearing Problem 8.1 towards the PA and AS cases, by seeking solutions of(13) with | T | replaced by | H : S | for H an almost simple group with socle T and S a maximal subgroupof H (compare Hypothesis 6.1, which reduces to the AS case if r is taken to be 1). However, solutionsof (13) seem to be rather more common in this setting, and so other methods are needed to rule outcases where solutions arise. For example, if we take H = T = McL (the McLaughlin sporadic simplegroup) then there are five (classes of) maximal subgroups S of H for which | H : S | = ( s + 1)( st + 1) IMPLE GROUPS, PRODUCT ACTIONS, AND GENERALISED QUADRANGLES 19 has an ‘admissible’ solution: four maximal subgroups of order 40320, which yield ( s, t ) = (296 , (3), for which ( s, t ) = (24 , We conclude with a brief discussion of theTW case. Let N = T × · · · × T r , where T ∼ = · · · ∼ = T r ∼ = T for some non-Abelian finite simplegroup T . A primitive permutation group G Sym(Ω) of type TW is a semidirect product N ⋊ P with socle N acting regularly by right mutlitplication, and P Sym r acting by conjugation in such away that T , . . . , T r are permuted transitively. Certain other rather complicated conditions must alsohold [ ], and in particular T must be a section of P . If we intend to apply Theorem 1.1 to classifythe generalised quadrangles with a point-primitive collineation group of TW type, then we will need‘good’ lower bounds for fixities of primitive TW-type groups. Liebeck and Shalev [ , Section 4] showthat, for every T and r , the fixity of G is at least | T | r/ . Although this is very far away from the4 / /
5, so that we could at least rule out some of thesubgeometries listed in Lemma 2.2 and then perhaps use the underlying point-regular group to saymore. In [ , Section 4], Liebeck and Shalev consider an involution x ∈ P (which must exist because T is a section of P and | T | is even) and observe that x induces a permutation of { T , . . . , T r } that fixesat least | T | ca + b elements of Ω ≡ T r , where the induced permutation has cycle structure (1 a , b ) and every involution g ∈ Aut( T ) satisfies | C T ( g ) | > | T | c . By [ , Proposition 2.4], we can take c = 1 / T , and so because a/ b > ( r − / / r/
3, it follows that x fixes at least | T | r/ elements. Now, c can certainly be taken larger than 1 / T (though presumably never as large as 4 / c > / ca + b ismaximised when b = 1 (else it is maximised when a = 0). Hence, roughly speaking, if c happens tobe somewhat large (for a given T ) and we happen to be able to guarantee that x can be chosen with b quite small, then we might have a useful bound on the fixity of G to work with. Bounds on c cancertainly be determined on a case-by-case basis from standard results about involution centralisers,but in light of the rather involved necessary and sufficient conditions for P to be a maximal subgroupof G , it is not clear to us what can be said about the cycle structure of permutations of { T , . . . , T r } induced by involutions in P . We therefore pose the following (somewhat vaguely worded) problem. Problem . Under what conditions can a primitive permutation group of type TW and degree d be guaranteed to have large fixity, where by “large” we mean, say, d / or more? Appendix A. Additional proofsProof of Proposition 2.4.
Suppose that we are not in case (iii). Then, by the final assertion ofLemma 2.2, we have |P θ | ( s + 1)( t + 1), and we argue as in the proof of Theorem 1.3. We mustshow that we are either in case (ii), or that f ( s, t ) = (( s + 1)( st + 1)) / − ( s + 1)( t + 1) is positive.We have ∂f∂t ( s, t ) = ( s + 1)(7 s − h ( s, t )) /h ( s, t ), where h ( s, t ) = 9(( s + 1)( st + 1)) / . If s >
13 then(using also 2 t s ) we have h ( s, t ) s ) / ( st ) / ) / s / , and so 7 s − h ( s, t ) > s / (7 s / − ) / ). The right-hand side is positive if and only if s > ( ) ( ) ≈ . >
12, andso it follows that f ( s, t ) > s >
13, for all s / t s . If 4 s
12 then a direct calculationshows that f ( s, t ) > s / t s , so it remains to consider s ∈ { , } . If s = 2 then, bythe final paragraph of the proof of Theorem 1.3, either t = 2 and every non-identity collineation of Q fixes at most 7 < |P| / = 15 / ≈ .
22 points, or t = 4 and we are in case (ii). Finally, if s = 3 then3 / t , and a direct calculation shows that f (3 , t ) > / t
7. Moreover, Q cannothave order ( s, t ) = (3 ,
8) by Lemma 2.1(iii). If ( s, t ) = (3 ,
9) then Q is the elliptic quadric Q − (5 , , 5.3.2], and a FinInG [ ] calculation shows that (up to conjugacy) there is a unique non-identitycollineation θ fixing 40 > |P| / = 112 / ≈ .
25 points. Moreover, Q θ is a generalised quadrangle oforder (3 , Q fixes at most 16 < / points. (cid:3) Proof of Proposition 2.5.
Suppose that we are not in case (ii) or (iii). Then |P θ | ( s + 1)( t + 1) bythe final assertion of Lemma 2.2. We show that f ( s, t ) = (( s +1)( st +1)) / − ( s +1)( t +1) is positive.We have ∂f∂t ( s, t ) = ( s + 1)(94 s − h ( s, t )) /h ( s, t ), where h ( s, t ) = 125(( s + 1)( st + 1)) / . Let a =2 . × . Then s > a , so (using also 2 t s ) we have h ( s, t ) a +1 a s ) / ( a +12 a st ) / (2 a +1)( a +1)2 a ) / s / , and hence 94 s − h ( s, t ) > s / (94 s / − (2 a +1)( a +1)2 a ) / ). Theright-hand side is positive because s > a > ( ) ( (2 a +1)( a +1)2 a ) ≈ . × , and it followsthat f ( s, t ) > s > a , for all s / t s . (cid:3) Proof of Proposition 2.6.
Since s = t − < t + 3, the final assertion of Lemma 2.2 implies that |P θ | ( s + 1)( t + 1) = ( s + 1)( s + 3). The result follows upon comparing this with |P| = ( s + 1) . (cid:3) Proof of Corollary 5.7.
By Propositions 4.2 and 5.4, T is one of the Lie type groups in the secondcolumn of Table 2. However, by Proposition 5.6, we must also have | C T ( x ) | < | T | / for all x ∈ T \{ } .We use this to show that T cannot have type A ε , B = C , F or G .If T ∼ = G ( q ) then q > (2) is not simple), | T | = q ( q − q − x ∈ T with | C T ( x ) | = q | A ( q ) | = q ( q − / gcd(2 , q −
1) as in the proof of Lemma 3.3(ii).If q is even or q >
19 then | C T ( x ) | > | T | / , and if q ∈ { , , , , , , , } then there is nosolution of | T | = ( s + 1)( st + 1) satisfying s > t > T ∼ = F ( q ) then q = 2 n +1 with n > F (2) is not simple and F (2) ′ was treatedin Proposition 5.4), | T | = q ( q + 1)( q − q + 1)( q −
1) and, as in the proof of Lemma 3.3(ii),we can choose x ∈ T with | C T ( x ) | = q | B ( q ) | = q ( q + 1)( q − | C T ( x ) | > | T | / for all q . If T ∼ = PSp ( q ) ∼ = Ω ( q ) then q > (2) ∼ = Sym is not simple), | T | = q ( q − q − / gcd(2 , q − x ∈ T with | C T ( x ) | = q ( q − / gcd(2 , q −
1) as inthe proof of Lemma 3.4 yields | C T ( x ) | > | T | / for all q >
3. Finally, if T ∼ = PSL ε ( q ) (whereL + := L and L − := U), then | T | = q ( q − q − q − ε ) / gcd(4 , q − ε ) and we can choose x ∈ T with | C T ( x ) | = q | GL ε ( q ) | / gcd(4 , q − ε ) = q ( q − q − ε ) / gcd(4 , q − ε ) as in the proof ofLemma 3.4. This yields | C T ( x ) | > | T | / unless ǫ = + and q = 2, and in this case there is no solutionof | T | = ( s + 1)( st + 1) satisfying s > t > (cid:3) Proof of Corollary 5.9 (continued).
Now suppose that T ∼ = PSL εn +1 ( q ) with n ∈ { , } , andchoose x ∈ T as in the proof of Lemma 3.4, so that (1) holds. If n = 8 then | C T ( x ) | > | T | / for all q >
2, so Proposition 5.6 gives a contradiction. If n = 7 then | C T ( x ) | > | T | / for all q >
2, socondition (ii) must hold. That is, s . × , so(14) | T | = |P| < ( s + 1) (2 . × + 1) < . × . This implies that q
9, in which case there is no solution of | T | = ( s + 1)( st + 1) satisfying s > t > T ∼ = PSp ( q ) then, as per (2), | T | = q ( q − q − q − q − / gcd(2 , q −
1) and we canchoose x ∈ T with | C T ( x ) | = q ( q − q − q − / gcd(2 , q − | C T ( x ) | > | T | / for all q >
2, so again (14) must hold. This implies that q
53, in which case there is no solution of | T | = ( s + 1)( st + 1) satisfying s > t > T ∼ = Ω ( q ) then we can take x ∈ T as in the proof of Lemma 3.4, so that with | C T ( x ) | = | SO ± n ( q ) | = q ( q ± q − q − q −
1) as in (3). This yields | C T ( x ) | > | T | / for all q >
2, so (14) musthold, and we immediately have a contradiction because | Ω ( q ) | = | PSp ( q ) | .Finally, suppose that T ∼ = PΩ ± n ( q ) with n ∈ { , } , and choose x ∈ T as in the proof of Lemma 3.4.If n = 8 then by using (4) (for q odd) and (6) (for q even), one may check that | C T ( x ) | > | T | / for all q >
2. Hence, (14) must hold, and this implies that q ∈ { , } , in which case there is nosolution of | T | = ( s + 1)( st + 1) satisfying s > t > n = 7. Then (14) holds if and only if q
4, and in this case there is no solution of | T | = ( s + 1)( st + 1) satisfying s > t > q ∈ { , } .Therefore, we must have q >
5. However, in this case | C T ( x ) | > | T | / , so we have a contradiction.(To check this, note that | C T ( x ) | > c · q ( q − q − ( q − q − q −
1) where c = or according as q is even or odd.) (cid:3) References [1] L. Babai, “On the automorphism groups of strongly regular graphs II”,
J. Algebra (2015) 560–578.[2] R. W. Baddeley, “Primitive permutation groups with a regular non-abelian normal subgroup”,
Proc. London Math.Soc. (1993) 547–595. IMPLE GROUPS, PRODUCT ACTIONS, AND GENERALISED QUADRANGLES 21 [3] J. Bamberg, M. Giudici, J. Morris, G. F. Royle, and P. Spiga, “Generalised quadrangles with a group of automor-phisms acting primitively on points and lines”,
J. Combin. Theory Ser. A (2012) 1479–1499.[4] J. Bamberg, A. Betten, P. Cara, J. De Beule, M. Lavrauw, and M. Neunh¨offer,
FinInG — Finite Incidence Geometry ,Version 1.0, 2014.[5] J. Bamberg and M. Giudici, “Point regular groups of automorphisms of generalised quadrangles”,
J. Combin.Theory, Ser. A (2011) 1114–1128.[6] J. Bamberg, S. P. Glasby, T. Popiel, and C. E. Praeger, “Generalised quadrangles and transitive pseudo-hyperovals”,
J. Combin. Des. (2016) 151–164.[7] J. Bamberg, S. P. Glasby, T. Popiel, C. E. Praeger and C. Schneider, “Point-primitive generalised hexagons andoctagons”, J. Combin. Theory, Ser. A (2017), 186–204.[8] J. Bamberg, T. Popiel, and C. E. Praeger, “Point-primitive, line-transitive generalised quadrangles of holomorphtype”, to appear in
J. Group Theory , doi:10.1515/jgth-2016-0042.[9] T. C. Burness and M. Giudici,
Classical groups, derangements, and primes , Australian Mathematical Society LectureSeries , Cambridge University Press, Cambridge, 2016.[10] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of finite groups , Oxford UniversityPress, Oxford, 1985.[11] E. Covato,
The involution fixity of simple groups , PhD Thesis, University of Bristol, in preparation.[12] S. De Winter and K. Thas, “Generalized quadrangles with an abelian Singer group”,
Des. Codes Cryptogr. (2006)81–87.[13] W. Feit and J. G. Thompson, “Solvability of groups of odd order”, Pacific J. Math. (1963) 775–1029.[14] The GAP Group, GAP — Groups, Algorithms, and Programming , Version 4.7.8, 2015.[15] D. Ghinelli, “Regular groups on generalized quadrangles and nonabelian difference sets with multiplier − Geom.Dedicata (1992) 165–174.[16] D. Gorenstein, R. Lyons and R. Solomon, The classification of the finite simple groups, number 3 , MathematicalSurveys and Monographs , American Mathematical Society, Providence, 1998.[17] D. Frohardt and K. Magaard, “Fixed point ratios in exceptional groups of rank at most two”, Comm. Algebra (2002) 571–602.[18] L. Morgan and T. Popiel, “Generalised polygons admitting a point-primitive almost simple group of Suzuki or Reetype”, Electronic J. Combin. (2016) P1.34.[19] C. E. Praeger, C.-H. Li and A. C. Niemeyer, “Finite transitive permutation groups and finite vertex-transitivegraphs”, in: G. Hahn and G. Sabidussi (Eds.), Graph symmetry , Kluwer, 1997, pp. 277–318.[20] M. W. Liebeck and G. M. Seitz,
Unipotent and nilpotent classes in simple algebraic groups and Lie algebras , Math-ematical Surveys and Monographs , American Mathematical Society, Providence, RI, 2012.[21] M. W. Liebeck and A. Shalev, “On fixed points of elements in primitive permutation groups”,
J. Algebra (2015)438–459.[22] S. E. Payne and J. A. Thas,
Finite generalized quadrangles , Pitman, London, 1984.[23] N. Spaltenstein, “Caract`eres unipotents de D ( F q )”, Comment. Math. Helvetici (1982) 676–691.[24] M. Suzuki, “On a class of doubly transitive groups”, Annals Math. (1962) 105–145.[25] J. Tits, “Sur la trialit´e et certains groupes qui s’en d´eduisent”, Inst. Hautes ´Etudes Sci. Publ. Math. (1959) 13–60.[26] S. Yoshiara, “A generalized quadrangle with an automorphism group acting regularly on the points”, European J.Combin. (2007) 653–664.[27] H. Van Maldeghem, Generalized polygons , Birkh´auser, Basel, 1998.
Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics,The University of Western Australia, 35 Stirling Highway, Crawley, W.A. 6009, Australia.Email: { john.bamberg, tomasz.popiel † , cheryl.praeger ‡ } @uwa.edu.au † Current address: School of Mathematical Sciences, Queen Mary University of London, Mile EndRoad, London E1 4NS, United Kingdom. ‡‡