Finite line-transitive linear spaces: parameters and normal point-partitions

(de Gruyter 2003

Finite line-transitive linear spaces: parameters and normal point-partitions

Anne Delandtsheer, Alice C. Niemeyer and Cheryl E. Praeger*

(Communicated by T. Penttila)

Abstract.Until the 1990’s the only known finite linear spaces admitting line-transitive, point- imprimitive groups of automorphisms were Desarguesian projective planes and two linear spaces with 91 points and line size 6. In 1992 a new family of 467 such spaces was constructed, all having 729 points and line size 8. These were shown to be the only linear spaces attaining an upper bound of Delandtsheer and Doyen on the number of points. Projective planes, and the linear spaces just mentioned on 91 or 729 points, are the only known examples of such spaces, and in all cases the line-transitive group has a non-trivial normal subgroup intransitive on points. The orbits of this normal subgroup form a partition of the point set called a normal point-partition. We give a systematic analysis of finite line-transitive linear spaces with normal point-partitions. As well as the usual parameters of linear spaces there are extra parameters connected with the normal point-partition that a¤ect the structure of the linear space. Using this analysis we characterise the line-transitive linear spaces for which the values of various of these parameters are small. In particular we obtain a classification of all imprimitive line- transitive linear spaces that ‘nearly attain’ the Delandtsheer–Doyen upper bound.

Key words.Finite linear space, line-transitive automorphism group, imprimitive permutation group.

2000 Mathematics Subject Classification. 20C05

1 Introduction

Afinite linear spaceD¼ ðP;LÞconsists of a finite setP of points, together with a set L of distinguished subsets of P, called lines, such that any two points lie on exactly one line, and each line contains at least two points. Theautomorphism group AutD ofD is the subgroup of all permutations ofPwhich leave Linvariant. We shall be concerned with finite linear spacesDfor which AutDacts transitively onL, that is,Disline-transitive. In particular, for such linear spaces, the lines have a con-

* This paper forms part of an Australian Research Council funded project of the second and third authors.

stant size,ksay, and so the linear space is a 2-ðv;k;1Þblock design, wherev¼ jPj.

We assume throughout the paper thatDhas more than one line, that isk<v.

It is well-known that in a 2-ðv;k;1Þdesign line-transitivity forces point-transitivity and that the two transitivities imply each other in finite projective planes. Over the last 30 years, various su‰cient conditions for point-primitivity of linear spaces have been studied, with the hope that ‘‘most’’ line-transitive linear spaces would prove to be point-primitive. Dembowski, in his book ‘‘Finite Geometries’’ [11], asked whether a line-primitive collineation group of a finite projective plane is necessarily point- primitive, and it took almost 20 years until this question was answered in the a‰r- mative by Kantor [14], using the classification of primitive permutation groups of odd degree (and hence relying on the finite simple group classification). However the question of whether line-primitivity implies point-primitivity for finite linear spaces is still open. Similar problems have been raised in more general contexts, for example, for incidence structures whose incidence matrices have maximal rank (see [19]). Hig- man and McLaughlin [13] proved that (point, line)-flag transitivity also implies point- primitivity in a 2-ðv;k;1Þdesign.

However line-transitivity alone does not imply point-primitivity for 2-ðv;k;1Þde- signs, and in this paper we shall studyimprimitive pairsðD;GÞ, whereDis a 2-ðv;k;1Þ design admitting a line-transitive but point-imprimitive subgroupGof AutD. ThusG leaves invariant a partitionCof the point set with classes of sizec, where 1<c<v, in the sense that for gAG and a classCACthe image C^g¼ fa^gjaACg is also a class of C. Examples of imprimitive pairs are provided by the Desarguesian projec- tive planes on a non-prime number of points, taking the groupGto be a cyclic Singer group. The only other known imprimitive pairs involve two 2-ð91;6;1Þdesigns found by Mills [16] and Colbourn & Colbourn [8], and 467 examples (up to isomorphism) which are 2-ð729;8;1Þdesigns (see [17]). The line-transitive 2-ð91;6;1Þdesigns were studied in [3]; the second one was named after McCalla by its discoverer Colbourn (Charlie Colbourn, private communication). In all these examples v and k are co- prime, and we are tempted to conjecture that this may be true in general.

Delandtsheer and Doyen [10] proved that the number of imprimitive pairs is bounded above by a function of the line size k, by showing that vc ^k₂ 12

. The 467 examples constructed in [17] were proved in [2] and [18] to be the only imprimitive pairs for whichvattains this upper bound, and they were found in the course of investigating this extreme case. The methodology involved a detailed group theoretic analysis to identify the possibilities for AutD, followed by a sophisticated computer search for the designs. The success of this classification demonstrated the importance of understanding the structure of line-transitive automorphism groups of such designs.

The aims of this paper are two-fold. Firstly we investigate some combinatorial con- sequences of having two structures on the point set left invariant by the automor- phism group, namely the set of lines and the point-partition. Secondly we study the structure of the line-transitive, point-imprimitive automorphism group. This enables us to characterise line-transitive linear spaces for which various of these parameters are small, see Theorem 1.2. Throughout the paper we will assume that the following hypothesis and notation hold.

Hypothesis 1.ðD;GÞis an imprimitive pair, that is,D¼ ðP;LÞis a 2-ðv;k;1Þdesign with point-setPand line-setL, andGcAutDis transitive onLand is imprimitive on P, leaving invariant a non-trivial point-partitionC, where Cnon-trivial means that bothd ¼ jCj>1 and the class sizec¼ jCj>1, forCAC. We assume thatDis non-trivial in the sense that 3ckcv3. By [10], the parametersc;d have the fol- lowing form

c¼

k

2 x

y and d ¼

k 2 y

x ; ð1Þ

where x;y are positive integers called the Delandtsheer–Doyen parameters. The integer x is the number of inner pairs on a line LAL, that is, the number of unordered pairs ofLwhich lie in the same class ofC. We denote the class ofCcon- taining a pointabyCðaÞ.

Our first aim is to investigate some of the rich combinatorial structure on the point setPgenerated by interactions between the lines ofLand the classes ofC. We define in Section 2 several parameters which help to measure these line-class inter- actions. The generic name we give to these parameters isintersection parameters, and we prove two technical propositions about them. These results are then used in Sec- tion 3 to obtain upper bounds on the sizes of the line-class intersections in terms of candx. For smallc, or smallx, it is possible to determine all possibilities for the set I0of non-zero line-class intersection sizes. In Tables 2 and 3 we give this information for cc12 and xc6 respectively. The values taken by the intersection parameters in the known examples of line-transitive, point-imprimitive linear spaces, apart from projective planes, are a small subset of these possible values and are recorded in Table 4. In particular, in all the known examples that are not projective planes, the Delandtsheer–Doyen parameterxis at most 2, and these tables of possibilities raise the still open question as to whether there may be further imprimitive pairsðD;GÞ corresponding to some of the other entries.

Question 1.Are there imprimitive pairsðD;GÞcorresponding to any of the parameter values in Table 3 other than the parameter values in Table 4?

The elementary arguments in Sections 2 and 3 show that minfc;dgd3 and that if eitherc¼3 ord ¼3, thenDmust be a projective plane (see Corollary 2.2 and Cor- ollary 2.5). However we do not even know if there exists an imprimitive pair with class sizec¼4 or withd ¼4 classes. For such an example one of the Delandtsheer–

Doyen parameters must equal 1, and it follows from [4, Lemma 8] that ifc¼4 then k;vmust satisfyk¼8hþ2,v¼4ð24h²þ9hþ1Þfor some positive integerh.

Question 2.Is there an imprimitive pairðD;GÞfor which minfc;dg ¼4?

Our inability to answer this question using elementary methods is rather dis- appointing, but more group-theoretic methods can provide extra information. A

recent result of Camina and the third author [7] implies that, for an imprimitive pair ðD;GÞ, if every non-trivial normal subgroup of G is transitive on the points of D, that is, ifGisquasiprimitiveon P, thenGisalmost simple, that isTcGcAutðTÞ for some non-abelian simple groupT. This suggests that it may be useful to consider separately those imprimitive pairs ðD;GÞ for which G is quasiprimitive on points, from those for which G is not quasiprimitive on points. Apart possibly from pro- jective planes, there are no known imprimitive pairs ðD;GÞ for which G is point- quasiprimitive, and we have chosen to focus in this paper on the case whereGis not point-quasiprimitive. Nevertheless, the question of existence of imprimitive pairs with a point-quasiprimitive group is of great interest and importance. It has recently been shown in [5] that no imprimitive pairs exist in the case whereGis quasiprimitive and is a finite alternating or symmetric group.

Question 3. Does there exist an imprimitive pair ðD;GÞ, with D not a projective plane, for whichGis point-quasiprimitive (and hence is almost simple)?

For any transitive group action, the set of orbits of a normal subgroup is an invariant partition of the point set. Such invariant partitions for the action of a group Gare calledG-normal partitions, and it follows from the definition of quasiprimitivity that a permutation groupG on a setP is quasiprimitive if and only if the only G- normal partitions are the trivial partitions (that is, the partition consisting of single- tons, and the partition consisting of the single class P). IfðD;GÞis an imprimitive pair with G not quasiprimitive on P, then we may take C to be a non-trivial G- normal partition ofP. It turns out that much stronger information can be obtained in this case than in the general case. In Section 4 we begin an investigation of the structure of line-transitive groups which preserve a non-trivial normal point- partition. If the partition C in Hypothesis 1 is the set of point orbits of a normal subgroupKofG, then by [6, Theorem 1] we know thatKacts faithfully on each of the classes ofC. SinceKis normal inGandGis line-transitive, allK-orbits inLhave the same length, sayb_K. In Proposition 4.1 we make a few observations aboutb_K, depending on both the form of the setI₀ of non-zero line-class intersection numbers (as defined in Section 2), and the relationships between the permutation groupsK^C induced on the classes CAC. Our results are strongest in the special case where I₀¼ f1;2g. These are given in Theorem 1.1 which is proved after Proposition 4.1. We note that, for all the known line-transitive, point-imprimitive linear spaces apart from projective planes, the set I₀ has this form (see Table 4). Also,I₀ must be f1;2g if xc2 (see Table 3).

Theorem 1.1.LetðD;GÞbe an imprimitive pair satisfying Hypothesis1withCthe set of K-orbits and K the kernel of G onC.Suppose also that I0 ¼ f1;2g.Then c is odd,all K-orbits on lines have length c,and KaðaAPÞis an elementary abelian2-group with all point orbits of length at most 2. Also, every minimal normal subgroup of G con- tained in K is elementary abelian of odd order dividing c.

Most of the arguments used in the proof have an elementary combinatorial and

group theoretic basis. However, for the last assertion we need to use the classification by Walter [20] of the finite simple groups with abelian Sylow 2-subgroups. We derive several consequences of this result in the cases where eithercorxis small in Theorem 1.2 below, the proof of which follows from Lemma 4.2.

Theorem 1.2.LetðD;GÞbe an imprimitive pair satisfying Hypothesis1withCthe set of K-orbits and K the kernel of G onC.Then the following all hold.

(a) If cc6then either c¼5,K¼Z₅ and I₀ ¼ f1;2g,or c¼3andD is a projective plane.

(b) If xc2,then either K is elementary abelian of order c¼p^a,for some odd prime p and ad1,orDsatisfies line1or2of Table1.

(c) If x¼1then c¼p^afor some odd prime p and integer ad1,and ydðc3Þ=2if p>3,or ydðc9Þ=18if p¼3.

Remark 1.3.(1) In Theorem 1.2(a), ifDis not a projective plane then it follows from Proposition 2.4(vi) thatDhas 2v¼10d lines withdodd, andx¼1, y¼ ðd1Þ=4.

No examples are known.

(2) The casex¼1 in Part (b) was proved in [18, Theorem 1.1].

Theorem 1.2 enables us to obtain a complete classification in the case wherex¼1, yc2. This is a generalisation of the classification in [18] for the casex¼y¼1 in the case where there is a normal point partition. The proof given below involved an exhaustive computer search and we are grateful to Anton Betten for carrying out this search.

Theorem 1.4.LetðD;GÞbe an imprimitive pair satisfying Hypothesis1withCthe set of K-orbits and K the kernel of G onC.If x¼1and yc2thenDsatisfies line3or4 of Table1.

Proof of Theorem 1.4. By Lemma 4.2, either D satisfies line 3 or 4 of Table 1 as claimed, or there is one further possibility, namely ðx;y;c;d;kÞ ¼ ð1;2;27;53;11Þ, the group Ghas a point-regular normal subgroup R¼Z53Z₃³, and Ghas a line- regular normal subgroupRZ13 of index dividing 4. (A permutation group is called regular if it is transitive and the only element fixing a point is the identity.) More- over a subgroup of order 13 acts non-trivially on each of the Sylow subgroups of

Table 1. Results for Theorems 1.2 and 1.4

x y c d k Comments

1 3 3 7 5 D¼PGð2;4Þ,K¼S₃,G¼KH, whereH¼Z₇orZ₇Z₃ 2 1 13 7 6 Colbourn design,K¼D₂₆,G¼Z₉₁Z_e, wheree¼6 or 12.

1 1 27 27 8 N²OP²designs [17]

1 2 7 13 6 Colbourn and Mills designs [8], [16]

R. An exhaustive computer search similar to that described in [1] was conducted for line-transitive 2-ð1431;11;1Þdesigns admitting a line-regular group of the form ðZ53Z₃³Þ Z₁₃. The search showed that no such designs exist.

2 Intersection parameters

In this section we introduce several parameters to describe the interaction between the lines of D and the partition C. We refer to these parameters generically as the intersection parametersforD;G;C. We denote the number of lines ofD byb. For a given point a, the remaining pointsPnfag are partitioned into disjoint sets of size k1 by the lines through a. Thus the number r of lines through each point is ðv1Þ=ðk1Þ. Counting incident point-line pairs we havebk¼vr. These equations, together with the well-known Fisher Inequality, namelybdv, are the basic relations between the parameters for any 2-ðv;k;1Þdesign.

We derive further equations related to theG-invariant partitionC. Note that since rdivides v1, we have gcdðr;vÞ ¼1. For an integer n we call n^ðrÞ¼gcdðn;rÞand n^ðvÞ¼gcdðn;vÞ the r-part and v-part of n, respectively. For a line L and a class C we call k_L;C:¼ jLVCj the line-class intersection number for C and L and, for 0cick, we say thatCandLarei-incidentifk_L;C¼i. The number of classes that arei-incident with some lineLis

d_L;i¼d_i¼ jfCACjk_L;C¼igj:

As G is line-transitive this number is independent of L, and hence is denoted di. Similarly the number of lines that arei-incident to some classCis

bC;i¼bi¼ jfLALjkL;C¼igj:

AsGis transitive onCthis number also is independent ofC. LetIbe the set of line- class intersection numbers, that is,

I:¼ fkL;CjLAL;CACg ð2Þ

and let I₀¼Inf0g be the set of all non-zero line-class intersection numbers. Our starting point is the following result which follows implicitly from the proof of Hig- man and McLaughlin in [13] (see also [9]).

Proposition 2.1(Higman and McLaughlin).For an imprimitive pairðD;GÞ,there are at least two distinct non-zero line-class intersection numbers,that isjI0jd2.

This proposition has several useful corollaries.

Corollary 2.2.IfðD;GÞis an imprimitive pair,then (a) a class does not contain a line;

(b) a line does not contain a class;

(c) the class size cd3.

Proof. If a line is contained in a class, thenI₀¼ fkg, which contradicts Proposition 2.1. Hence (a) holds. If some line contains a class of C, then every line contains a class ofCsinceGis line-transitive. However each class ofCis contained in at most one line as Dis a linear space. Hence bcjCj ¼d <v, which is a contradiction. In particular, this means thatc02.

We extend the meaning of i-incidence for lines and classes, by defining a pointa and a line L to be i-incident if aAL and k_L;CðaÞ¼i, where CðaÞ is the class ofC containing a. The number of points that are i-incident to some line Lcan be com- puted as

id_i¼ jfaAPjaAL;k_L;CðaÞ¼igj:

The number of lines that arei-incident to the pointais called thei-degreeofaand is denoted byra;i. As Gis transitive onP and preservesi-incidence, the numberra;iis independent ofaand is usually denotedr_i. Thus

ra;i¼ri¼ jfLALjaAL;k_L;CðaÞ¼igj;

and by convention,r₀¼0.

The line-sizekand the total number of linesb¼vr=kcan be factorised into theirv andrparts as follows:

k^ðrÞ¼gcdðk;rÞ ¼gcdðk;v1Þ; k^ðvÞ¼gcdðk;vÞ;

b^ðrÞ¼gcdðb;rÞ ¼gcdðb;v1Þ; b^ðvÞ¼gcdðb;vÞ:

Then, sincebk¼vrand gcdðv;rÞ ¼1,

k¼k^ðvÞk^ðrÞ and b¼b^ðvÞb^ðrÞ: ð3Þ Using again gcdðv;rÞ ¼1, we deduce fromvr¼bk¼b^ðvÞb^ðrÞk^ðvÞk^ðrÞthat

v¼k^ðvÞb^ðvÞ and r¼k^ðrÞb^ðrÞ: ð4Þ Next we record some information about the configuration induced on a classCby i-incidence, whereiAI,id2. Recall that such aniexists by Proposition 2.1.

Proposition 2.3.Let ðD;GÞbe an imprimitive pair.Let C AC,and for0cick, let SðiÞ ¼ fCVLjLAL;jCVLj ¼ig(SðiÞmay be empty).Then

(i) bdi¼dbi and ibi¼cri,and hence bdii¼vri. (ii) b_i⁰:¼^cdi

k^ðvÞ¼ ^bi

b^ðrÞand r_i⁰:¼ ^idi

k^ðvÞ¼ ^ri

b^ðrÞ are integers;moreover,cr_i⁰¼ib_i⁰.

(iii) If iAI and id2, then ðC;SðiÞÞ is a 1-design admitting GC acting point- transitively, with bi blocks, c points, block size i, and ri blocks on each point.

Moreover,

(a) r_id2,and if r_i¼2then i¼2andDis a projective plane;

(b) cdr_iði1Þ þ1,with equality if and only if I₀¼ f1;ig,which in turn holds if and only ifðC;SðiÞÞis a2-ðc;i;1Þ-design.

Proof. Counting the number of i-incident class-line pairs ðC;LÞin two ways yields bdi¼dbi. For a fixed class Ccounting the number ofi-incident point-line pairs ða;LÞ withaAC yields the second equality of (i). Then using the first equality we havebd_ii¼db_ii¼dcr_i¼vr_i.

Using Part (i), we have b^ðrÞd_i¼ ðb=b^ðvÞÞ ðdbi=bÞ ¼ ðdbiÞ=b^ðvÞ¼ ðvbiÞ=ðb^ðvÞcÞ ¼ ðk^ðvÞb_iÞ=c. Hence b^ðrÞcd_i¼k^ðvÞb_i, and as gcdðk^ðvÞ;b^ðrÞÞ ¼1, it follows that b_i⁰:¼ cd_i=k^ðvÞ¼b_i=b^ðrÞis an integer. By Part (i),b^ðvÞb^ðrÞd_ii¼b^ðvÞk^ðvÞr_iand sob^ðrÞd_ii¼k^ðvÞr_i. Since gcdðk^ðvÞ;b^ðrÞÞ ¼1, it follows thatr_i⁰:¼id_i=k^ðvÞ¼r_i=b^ðrÞis an integer. By defini- tion,cr_i⁰¼ib_i⁰. Thus (ii) is proved.

Now suppose thatiAIandid2. The fact thatðC;SðiÞÞis a 1-design follows from the discussion preceding the proposition. LetLALbe such thatS:¼CVLASðiÞ.

Ifri¼1 thenSis a block of imprimitivity forGCinC, and henceSis also a block of imprimitivity forGinP, contradicting Corollary 2.2(b). Hencerid2. Suppose next that ri¼2. Then by Part (i), bi¼2c=icc. SinceLis the unique line containingS, GS fixes L setwise, and hence GScGL so b¼ jG:GLjcjG:GSj ¼djGC:GSjc dbicdc¼vcb. Henceb¼v, and so D is a projective plane. Also we must have bi¼c, soi¼2 by (i). Thus (iii)(a) is proved.

Let aAC. We count incident point-line pairs ðb;LÞ for which bACnfag. As two points determine a line there are c1 choices for b and 1 choice forL. On the other hand, for eachhd1, there arer_hchoices for lines that containaand meetCin h1 other points. Hencec1¼P

hr_hðh1Þ. It follows thatcdr_iði1Þ þ1 with equality if and only if IJf0;1;ig. This holds if and only if any line intersecting a classCin at least two points has exactlyipoints inC, or equivalentlyDinduces the structure of a 2-ðc;i;1Þ-design onC.

In the course of the proof above, we proved the equalityc1¼P

hrhðh1Þ. We state this formally below, together with other equalities obtained by averaging over the setI. Recall that by conventionr0 ¼0.

Proposition 2.4.LetðD;GÞbe an imprimitive pair satisfying Hypothesis1.

(i) k¼Pk

i¼0idi¼P

CACkL;C,for LAL,and d ¼Pk i¼0di; (ii) r¼Pk

i¼0riand b¼Pk i¼0bi; (iii) rc¼P

iAIb_ii¼P

LALk_L;C,for CAC;

(iv) c1¼P

iAIr_iði1Þ;

(v) x¼P

iAI i

2 d_i¼P

iAI ði1Þ

2 r_i⁰k^ðvÞ¼^ðc1Þk_2bðrÞ^ðvÞ,with r_i⁰as in Proposition2.3.

(vi) ðd1Þx¼ ðc1Þy,c1¼2xb=v and d1¼2yb=v.

Proof.The equalities in (i) and (ii) follow from the definitions of thed_i,k_L;C,r_iand

b_i. We obtain (iii) by counting incident point-line pairsða;LÞforaAC. The equality in (iv) was proved in the proof of Proposition 2.3. For (v), the number x of inner pairs on a line is P

iAI i

2 d_i, which, by Proposition 2.3(ii) is P

iAIr_i⁰k^ðvÞði1Þ=2¼ P

iAIði1Þri

k^ðvÞ=2b^ðrÞ, and by (iv) this equalsðc1Þk^ðvÞ=2b^ðrÞ. From Hypothesis 1 we have

ðd1Þx¼

k 2 y

x 1

!

x¼ ^k₂ yx¼ ðc1Þydc1;

which is the first equality in (vi). Sincek^ðvÞ=b^ðrÞ¼v=b, the second part of (vi) follows from (v). The third part is then immediate.

We can use these relationships between the parameters to obtain a lower bound on the numberdof classes and on their sizec, and in particular to prove that bothcand dare at least 3.

Corollary 2.5.LetðD;GÞbe an imprimitive pair satisfying Hypothesis1.Then c and d are both at least1þ2b=v.In particular,cd3,dd3,and if c¼3or d ¼3thenDis a projective plane.

Proof.The first inequalities follow from Proposition 2.4(vi) and the fact thatxand y are positive integers. The other assertions are then consequences of Fisher’s Inequal- ity, namelybdvwith equality if and only ifDis a projective plane.

Refinement of the intersection parameters to orbit parameters. Let ðD;GÞ be an imprimitive pair as in Hypothesis 1. Then there is an equivalence relation onLC such that two pairs ðL;CÞ;ðL⁰;C⁰ÞALC are equivalent whenever jLVCj ¼ jL⁰VC⁰j. The partition ofLCdetermined by this equivalence relation has equi- valence classes indexed by the setIof line-class intersection numbers defined in (2).

Moreover the group G fixes each equivalence class setwise, and so this partition is refined by the partition of LC into G-orbits. For each iAI let O_ði;1Þ;O_ði;2Þ;. . . denote theG-orbits oni-incident line-class pairs, and letIHIN₀ denote the set of indices ði;jÞ indexing all G-orbits in LC. For a line L and a classC we let h_L;C:¼ ði;jÞif and only ifðL;CÞAO_ði;jÞand we say thatCandLareði;jÞ-incident if h_L;C¼ ði;jÞ. Hence I¼ fhL;CjLAL;CACg and we can generalize the above definitions ofd_i,b_iandr_itod_ði;jÞ,b_ði;jÞandr_ði;jÞ, so that

X

j

dði;jÞ¼di; X

j

bði;jÞ¼bi; X

j

rði;jÞ¼ri:

It is then straightforward to refine Propositions 2.3 and 2.4 using these orbit parameters.

We finish this section by examining the orbit lengths of a point stabiliser in its actions on lines and on the partitionC. The results below strengthen Proposition 2.3.

We denote byG^Cthe permutation group induced byGon the partitionC. We note that part of this result can be found in [15, Proposition 3.3 and Corollary 3.2].

Proposition 2.6. Let ðD;GÞ be an imprimitive pair satisfying Hypothesis 1, and let aAP.Then the parameter b^ðrÞdivides the length of

(a) every G_a-orbit on the set of lines througha;

(b) every Ga-orbit onPnfag,and in particular on CðaÞnfag;and

(c) every Ga-orbit onCnfCðaÞg,and hence also every GC-orbit onCnfCg.

Proof.Suppose thata lies on the lineL, and letabe the length of theGa-orbit con- tainingLinL. Thenva¼ jG:Gaj jGa:Ga;Lj ¼ jG:Ga;Ljis divisible byjG:GLj ¼ b. Sinceb^ðrÞdividesband is coprime tov, it follows thatb^ðrÞdividesa. Thus Part (a) is proved.

LetGbe aG_a-orbit inPnfag. The lines throughainduce a partition ofG. Any two lines in the sameG_a-orbit intersectGin the same number of points, sojGjis divisible by the length of theG_a-orbit of any line throughaintersectingG. Thus by Part (a), b^ðrÞdividesjGj. This holds in particular for theG_a-orbits onCðaÞnfag, so Part (b) is proved.

LetDbe aG_a-invariant subset ofCnfCðaÞg, lete¼ jDj, and letPDbe the union of the classes ofCthat are contained inD. We claim thatb^ðrÞ dividese. Note that Part (c) follows from this claim. LetLD be the subset ofLconsisting of lines containing aand intersecting some class inDnon-trivially. SinceDisGa-invariant, so alsoLDis Ga-invariant. LetL1;. . .;Lt be theGa-orbits inLD. By Part (a), eachjLijis divisible byb^ðrÞ. For eachi, chooseLiALi. Note that, for eachbAPD, the unique line con- tainingaandblies inLD. Counting the number of pairsðb;LÞwherebAPD,LALD, andbAL, we find

ec¼X^t

i¼1

jLij jLiVPDj:

Sinceb^ðrÞdivides eachjLij, it follows thatb^ðrÞdividesec, and sincev¼cd is coprime tob^ðrÞwe conclude thatb^ðrÞdividese.

3 Bounds on line-class intersection numbers

LetD;G;Cbe as in Hypothesis 1, and letIbe the set of line-class intersection num- bers. Set

i_max¼maxfijiAIg: ð5Þ

By Corollary 2.2,imax is strictly less than the class sizec. The equalities of Proposi- tion 2.4 allow us to reduce this upper bound to approximatelypffiffiffic

.

Lemma 3.1.i_max<pffiffiffic

þ1=2,and i_max< ffiffiffiffiffiffi p2x

þ1.

Proof.By Proposition 2.4(iv) and Proposition 2.3,

c1¼X

iAI

r_iði1Þ ¼b v

X

iAI

d_iiði1Þdi_maxðimax1Þ;

and the first inequality follows. By Proposition 2.4(v), 2xdimaxðimax1Þ, and we obtain the second inequality.

This result suggests that it may be possible to obtain a limited number of feasible sets of intersection parameters if either or both ofcandxare bounded. Such infor- mation may lead to the discovery of new block-transitive, point-imprimitive linear spaces, and thereby lead to a better understanding of such linear spaces. Accordingly we determine the possibilities forIand the parametersd_ifor small values ofcandx.

We note that upper bounds onxalone do not imply upper bounds forc.

Lemma 3.2.For cc12,the possibilities for I₀,c,x and the d_iare given in Table2.For xc6the possibilities for I₀,x and the d_iare given in Table3.

Proof.By Proposition 2.1,jI0jd2. By Propositions 2.3(i) and 2.4(iv) and (v),

c1¼X

iAI

riði1Þ ¼b v

X

iAI

diiði1Þ ¼2bx v : Also, by Lemma 3.1,i_max<pffiffiffic

þ1=2 andi_max< ffiffiffiffiffiffi p2x

þ1. These inequalities imply that, forcc12, we havei_maxc3, and also forxc6 we havei_maxc4.

Table 2. Intersection parameters for 3ccc12

I₀ c d_i b=v x

f1;2g 3ccc12 1cd2cðc1Þ=2 ðc1Þ=2d2 d2

f1;3g 7ccc12 d₃¼1 ðc1Þ=6 3 f2;3gorf1;2;3g 9ccc12 1cd₂cðc7Þ=2,d₃¼1 ðc1Þ=ð2d2þ6Þ d₂þ3

Table 3. Intersection parameters for 1cxc6

I₀ d_i b=v x

f1;2g 1cd₂c6 ðc1Þ=2d2 d₂

f1;3g d₃¼1 or 2 ðc1Þ=6d3 3d₃

f1;4g d₄¼1 ðc1Þ=12 6

f2;3gorf1;2;3g 1cd₂c3,d₃¼1 ðc1Þ=2ðd2þ3Þ d₂þ3

IfI₀¼ f1;2g, then the numberxof inner pairs isd₂, and the displayed equations above give c1¼ ðb=vÞ 2d₂d2d₂, and hence the first line of Table 2 holds if cc12, and the first line of Table 3 holds ifxc6. If 3ccc6, then i_max¼2, and henceI₀¼ f1;2g, and we have the required result.

Thus we may assume that cd7 and that i_maxd3, that is, I₀0f1;2g. If I0¼ f1;3g, then x¼3d3, and c1¼ ðb=vÞ 6d3d6d3. Hence the second line of Table 3 holds ifxc6. If 7ccc12, thenimax¼3, andc1¼ ðb=vÞð2d2þ6d3Þd 2d2þ6d3d2d2þ6, so either d2 ¼0, d3¼1 and I0 ¼ f1;3g, or we have 1cd2c ðc7Þ=2,d3¼1 andcd9. Hence ifc¼7 or 8 then the second line of Table 2 holds.

In summary, if either 7ccc8 orI0¼ f1;3g, then we have required result. If 9c cc12 andI00f1;2g andf1;3g, thenI0¼ f2;3gorf1;2;3g, and the third line of Table 2 holds.

Thus we may assume thatxc6 andI00f1;2gandf1;3g. Sinceimax< ffiffiffiffiffiffi p2x

þ1, we have 3cimaxc4 and 2x¼P

diiði1Þ. Ifimax¼4, this implies thatI0¼ f1;4g and d₄¼1, so x¼6; also c1¼ ðb=vÞP

d_iiði1Þ ¼12b=v, so the third line of Table 3 holds. Thus we may assume that i_max¼3. Then 12d2x¼2d₂þ6d₃ and both d₂ andd₃ are non-zero, so d₃ ¼1 and d₂¼x3 is 1, 2 or 3. Finallyc1¼ ðb=vÞP

d_iiði1Þ ¼ ðb=vÞð2d2þ6Þ, and hence the fourth line of Table 3 holds.

Comparable information about all the known line-transitive, point imprimitive linear spaces, apart from projective planes, is contained in Table 4. In both of the 2- ð91;6;1Þ designs there are two non-trivial invariant partitions, as in lines 1 and 2 of Table 4. For most of theN²OP² designs there is a unique non-trivial invariant partition, but for a few of these designs there are two such partitions, and for two of these designs there are 28 such partitions corresponding to the 28 parallel classes of lines in the a‰ne plane AGð2;27Þ. For all of the partitions corresponding to the N²OP²designs, the intersection parameters are as in line 3 of Table 4.

4 Normal partitions

Now we assume thatðD;GÞis an imprimitive pair satisfying Hypothesis 1 such that G has a non-trivial normal subgroupK which is intransitive on points. We further assume thatCis the set ofK-orbits inP, and thatKis the subgroup consisting of all elements ofGwhich fix every class ofCsetwise, that is,Kis the kernel of the action ofGonC. By [6, Theorem 1],Kacts faithfully on each class ofC, that is, for each CAC, the permutation groupK^Cinduced byKonCis isomorphic toK. SinceKis normal inGandGis line-transitive, it follows that allK-orbits inLhave the same

Table 4. Intersection parameters for known examples

I₀ c d x¼d₂ y b=v k Comments

f1;2g 7 13 1 2 3 6 Colbourn and Mills designs [8], [16]

f1;2g 13 7 2 1 3 6 Colbourn and Mills designs [8], [16]

f1;2g 27 27 1 1 13 8 N²OP²designs [17]

length, sayb_K. We begin by making a few simple observations aboutb_K, depending on both the form of the setIof line-class intersection numbers, and the relationships between the various permutation groupsK^C,CAC.

Proposition 4.1.LetðD;GÞbe an imprimitive pair satisfying Hypothesis1withCthe set of K-orbits and K the kernel of G on C. Let I0 be the set of non-zero line-class intersection numbers,and let bK be the length of the K-orbits on lines.

(a) Then b_K divides c.

(b) If1AI₀,then b_K¼c,where c is odd,and K_L¼K_afor any pointaand any line L such that LVCðaÞ ¼ fag.

Proof.Leta;bbe distinct points inP, letLdenote the unique line containingaandb, and letC⁰ACnfCðaÞg. Then K_abcK_Land henceb_K¼ jK:K_LjdividesjK:K_abj ¼ jK:Kaj jKa:Kabj ¼cjKa:Kabj. Thus bK divides c times the length of every Ka- orbit in Pnfag. Since Ka leaves CðaÞnfag invariant it follows that bK divides cjCðaÞnfagj ¼cðc1Þ, and since Ka leaves C⁰ invariant it follows that bK divides cjC⁰j ¼c². HencebKdivides gcdðcðc1Þ;c²Þ ¼c.

Suppose that 1AI0. Let aAP and LAL be such that LVCðaÞ ¼ fag. Then KLcKa, and hence c¼ jK:Kaj divides bK ¼ jK:KLj. By Part (a) it follows that bK¼cand henceKL¼Ka. Now letbACðaÞnfag and letL⁰ be the line containing aandb. ThenKfa;bgcKL⁰, and hencec¼bK¼ jK:KL⁰jdividesjK :Kfa;bgj. Thusc divides the length of everyK-orbit on unordered pairs from a classCAC, and soc dividescðc1Þ=2. It follows thatcis odd.

Our next task is to prove Theorem 1.1. We use the concept of permutational equivalence defined as follows. Suppose that a group H acts on sets X;Y. These actions, and also the corresponding permutation groups H^X;H^Y, are said to be permutationally isomorphic if there is a bijection c:X !Y and an automorphism jAAutðHÞsuch that, for allaAX,hAH, we haveððaÞcÞ^ðhÞj¼ ða^hÞc; ifj¼1 then the actions and permutation groups are said to be permutationally equivalent or simplyequivalent.

Proof of Theorem1.1. LetðD;GÞbe an imprimitive pair satisfying Hypothesis 1 with Cthe set ofK-orbits andKthe kernel ofGon C, and suppose that I0¼ f1;2g. Let p¼ fa;bgHC, and letLðpÞbe the line containingp. ThenLðpÞVC¼p, and hence Kp¼KLðpÞ. Then the Ka-orbit containing b has length jKa:Kabj ¼ jK:Kabj=c¼ jKp:Kabjc2. Thus allKa-orbits inChave length 1 or 2 and henceKaGK_a^C is an elementary abelian 2-group. Sincecis odd,Kamust therefore fix a point in each class ofC, and hence theK-actions on the classes ofCare all equivalent. Since for each C⁰ACthere existsa⁰AC⁰such thatKa¼Ka⁰, it follows that allKa-orbits inC⁰have length at most 2 also.

Suppose thatNis a minimal normal subgroup ofGcontained inK. ThenN¼T^t for some simple group T and some td1. Suppose that T is a non-abelian simple group. Then N_a¼NVK_a is an elementary abelian 2-group and all N_a-orbits in

P have length at most 2. Also, jN:N_aj ¼ jNKa:K_aj divides jK:K_aj ¼c, and hencejN:N_ajis odd. ThusN_a is a Sylow 2-subgroup ofN, and hence the Sylow 2- subgroups of T are elementary abelian. By a result of Walter [20], T is one of PSLð2;qÞ, q¼2ⁿd4 or q13;5ðmod 8Þ, Janko’s first group J₁, or ReeðqÞ⁰, q¼ 3^2aþ1d3. Since all N_a-orbits in P have length at most 2, it follows that distinct Sylow 2-subgroups of T intersect in a subgroup of index 2 of each. In the case of PSLð2;qÞ,q¼2ⁿd4, distinct Sylow 2-subgroups ofTintersect trivially, which is a contradiction. Similarly, in the other cases, a Sylow 2-subgroupSofTis contained in a subgroupA4, 2A5, or 2PSLð2;qÞ(and hence in 2A4), respectively, and it is easy to verify thatSintersects one of its conjugates in a subgroup of index 4. These contradictions imply thatT ¼Zp for some prime p. SinceN^Cis normal in the tran- sitive groupK^C it follows that allN-orbits in Chave the same size. In particular p dividesjCj ¼csopis odd and hencejNjis odd. Also, sinceKais a 2-group andKis faithful onC, it follows thatNVKa¼1 sojNjdividesc.

We complete this section by proving Theorem 1.2. This is achieved by proving the following lemma which also deduces the extra information needed for Theorem 1.4.

Lemma 4.2. LetðD;GÞbe an imprimitive pair satisfying Hypothesis1 withCthe set of K-orbits and K the kernel of G onC.Then all the assertions of Theorem1.2 hold.

Moreover, if x¼1 and yc2, then either D satisfies line 3 or 4 of Table 1, or ðx;y;c;d;kÞ ¼ ð1;2;27;53;11Þ,the group G has a point-regular normal subgroup R¼ Z53Z₃³, G has a line-regular normal subgroup RZ13 of index dividing 4, and a subgroup of G of order13acts non-trivially on each of the Sylow subgroups of R.

Proof. First we prove Theorem 1.2. Suppose that either cc6 or xc2. Then (see Tables 2 and 3)I0¼ f1;2g, and hence, by Theorem 1.1,cis odd, andKaðaAPÞis an elementary abelian 2-group which by [6, Theorem 1] acts faithfully onCðaÞ. First we show that Theorem 1.2(a) follows from Theorem 1.2(b). Suppose thatcc6 and that Theorem 1.2(b) is true. Then by Table 2, xcðc1Þ=2<3 so xc2. Since cc6, it follows from Theorem 1.2(b) that eitherDis PGð2;4Þ,K¼S₃andc¼3, or elseK¼Z_c andc¼3 or 5. If c¼3 thenD is a projective plane by Corollary 2.5.

Thus Theorem 1.2(a) follows.

Our next step is to prove Theorem 1.2(b). If x¼1 then by [18, Theorem 1.1], c¼p^afor some odd prime pand integerad1 and eitherK¼Z_p^a or line 1 of Table 1 holds. Thus Theorem 1.2(b) holds ifx¼1. Suppose now thatx¼2, and letCAC andaAC. First we prove that Cis a minimal block of imprimitivity forG. If this is not the case then there exists a proper subset BHC containing a such that Bis a block of imprimitivity forGandc0¼ jBjd2. By (1),c0¼ ^k₂ x0

=y0, for some positive integersx0;y0. SinceBis a block of imprimitivity forGCinCit follows that c0dividesc¼ ^k₂ 2

=y, and hence

k

2 x0dividescyy0¼y0 k 2 2

: ð6Þ

LetC0 ¼B^Gdenote the point-partition generated byB. ThenC0is a refinement ofC,

so every pair of points lying in the same block ofC0 is an inner pair forC. However, sinceGis line-transitive, every line contains at least one inner pair of points which do not lie in the same block ofC0. Hence 1cx₀<x¼2, sox₀¼1, and hence ^k₂ x₀ is relatively prime to ^k₂ 2. Therefore, by (6), ^k₂ x₀ divides y₀, but this means thatc₀c1, which is a contradiction. HenceCis a minimal block of imprimitivity.

Thus the stabiliserG_CofCACinduces a primitive groupG_C^ConC. It follows that a minimal normal subgroupMofGcontained inKmust be transitive and faithful on C. Then by Theorem 1.1 we deduce thatc¼ jMj ¼p^a for some odd prime p and integer ad1, M¼Z_p^a, K¼MKaðaAPÞ, and Ka is an elementary abelian 2- group acting faithfully onC. IfKa¼1 thenK¼Z_p^aand Theorem 1.2(b) holds, so we may assume thatK_a01. Arguing as in the proof of Theorem 1.1, the actions ofK on the blocks ofCare permutationally equivalent. ThusK_afixes the same number of points in each block of C. Since K^C is a normal subgroup of G_C^C, the set of fixed points of Ka in Cis a block of imprimitivity for G_C^C, and sinceG_C^C is primitive it follows thatKafixes exactly one point ofC. ThusKa fixes exactly one point of each block ofC. The setFof fixed points ofKais therefore a block of imprimitivity forG of sized, and theG-invariant partition F¼F^G it generates containscblocks. The integer yis the number ofF-inner pairs of points on each line. Letfb;ggbe an orbit ofKaof length 2, and letLbe the unique line containingfb;gg. SinceKafixesfb;gg, we have KacKL. By Theorem 1.1,jK:KLj ¼cand henceKa¼KL. SinceKa has a unique fixed point in each block ofC, it follows thatLconsists ofx¼2 orbits of K_aof length 2, andk4 points ofF, and by Proposition 2.1,k4d1. These latter k4 points all lie in the single blockFAF, while eachC-inner pair onLconsists of one point from each of two di¤erentF-blocks which are interchanged byK_a. Hence

y¼^k4₂

þd, whered¼0 or 2. Now by (1) and Corollary 2.5,

k

2 2¼cyd3y¼3 ^k4₂ þd

d3^k4₂ sokc9. Thus 5ckc9. Since y¼^k4₂

þddividescy¼ ^k₂ 2, and d¼0 or 2, we see from Table 5 thatðk;c;d;yÞis eitherð6;13;7;1Þorð5;4;4;2Þ. It follows from [4] that the latter case cannot arise, and from [3] and [4] that in the former caseDis the design discovered by Colbourn and Colbourn, and for this design there is a group Gfor which the subgroupK_a is cyclic of order 2. Thus line 2 of Table 1 holds, and Theorem 1.2(b) is proved.

Now we deal with Theorem 1.2(c), so suppose that x¼1. By Theorem 1.2(b), c¼p^a for some odd prime p and integerad1. We havec¼p^a¼ ^k₂ 1

=y, so 2yp^a¼k²k2¼ ðk2Þðkþ1Þ. If p>3, then, as gcdðk2;kþ1Þdivides 3, p^a

Table 5. Computations for the divisibility condition

k 9 8 7 6 5

k

2 2 34 26 19 13 8

k4 2

10 6 3 1 0

must divide eitherk2 orkþ1. Hencep^ackþ1¼ ðk2Þ þ3¼2yp^a=ðkþ1Þ þ 3c2yþ3, and so ydðc3Þ=2. Similarly, if p¼3 then 3^a1 divides one ofk2 or kþ1, and we obtain the required lower bound for y of Theorem 1.2(c). This completes the proof of Theorem 1.2.

To finish the proof of Lemma 4.2, we suppose that x¼1 and yc2. Then the inequalities of Theorem 1.2(c) alone show that cAf3;5;7;9;27g, and if c¼7 then y¼2. Next the fact that ^k₂ ¼1þcyimplies thatðy;c;kÞis one ofð1;5;4Þ,ð1;9;5Þ, ð1;27;8Þ,ð2;7;6Þ,ð2;27;11Þ. It was shown in [17] and [18] that the first two triples do not arise, and that in the case of the third triple, the examples are precisely the 467 examples constructed in [17] so line 3 of Table 1 holds. In the case of the fourth triple, it was shown in [4, Theorem 1] that the only designs which arise here are the ones of Mills and Colbourn so line 4 of Table 1 holds. So suppose we are in the last case.

By Theorem 1.2(b),Kis elementary abelian of orderc¼27. Alsod ¼ ^k₂ y¼53, and G=K acts faithfully and transitively on C of degree 53. Since 53 is prime, this action of G=K is primitive, and by [12, Table B.4], the only primitive groups of degree 53 are subgroups of AGLð1;53Þ and the alternating and symmetric groups A₅₃ and S₅₃. Since G is line-transitive and each line contains exactly x¼1 inner pair, it follows that the setwise stabiliserG_Cof a classCACmust be transitive on the pairs of points from C. If G=K ¼A₅₃ or S₅₃, then the group A₅₃ would centralise K, making it impossible for G_C to induce a permutation group on Ctransitive on unordered pairs. HenceG=KcAGLð1;53Þ. SincejAutKj ¼ jGLð3;3Þjis not divisi- ble by 53, it follows that Kis centralised by a subgroup of order 53, and hence G has a normal subgroupRGZ53K, andG=RcZ52. The groupRis transitive onC andRC¼Kis transitive onC, and it follows thatRis regular on points. SinceGCis transitive on the 2713 unordered pairs fromCit follows thatG=Rhas order divisi- ble by 13, and henceGhas a normal subgroupN¼RZ13 of index dividing 4. Now Nis transitive, and hence regular, on inner pairs, and henceNis regular on lines. We note that a subgroup of GC of order 13 must act non-trivially on both Sylow sub- groups ofR. This completes the proof.

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Received 19 December, 2000; revised 6 August, 2002

A. Delandtsheer, Faculty of Applied Sciences, CP165/11 Mathematics, Universite´ Libre de Bruxelles, Avenue F. D. Roosevelt 50, B 1050 Bruxelles, Belgium.

Email: [email protected]

A. C. Niemeyer, Ch. E. Praeger, Department of Mathematics & Statistics, University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia.

Email: {alice, praeger}@maths.uwa.edu.au