Odd order flag-transitive a‰ne planes of dimension three over their kernel

(de Gruyter 2003

Odd order flag-transitive a‰ne planes of dimension three over their kernel

Ronald D. Baker, C. Culbert*, Gary L. Ebert* and Keith E. Mellinger*

Dedicated to Adriano Barlotti on the occasion of his 80th birthday

Abstract.With the exception of Hering’s plane of order 27, all known odd order flag-transitive a‰ne planes are one of two types: admitting a cyclic transitive action on the line at infinity, or admitting a transitive action on the line at infinity with two equal-sized cyclic orbits. In this paper we show that when the dimension over the kernel for these planes is three, then the known examples are the only possibilities for either of these two types. Moreover, subject to a relatively mild gcd condition, one of these two actions must occur. Hence, subject to this gcd condition, all odd order three-dimensional flag-transitive a‰ne planes have been classified.

1 Introduction

Letq¼p^ebe an odd prime power. In [9] it is shown that if gcd¹₂ðqⁿþ1Þ;ne

¼1, then with the exception of Hering’s plane of order 27 any non-Desarguesian flag- transitive a‰ne plane of orderqⁿwhose kernel contains GFðqÞmust admit a (cyclic) Singer subgroup action which is either regular on the line at infinity or has two equal-sized orbits on the line at infinity. In the latter case these two orbits are joined by some other element in the translation complement, but there is no cyclic reg- ular action on the line at infinity. This result holds in all known odd order non- Desarguesian flag-transitive a‰ne planes, except Hering’s, whether or not the above gcd condition is satisfied. Flag-transitive a‰ne planes of the first type will be called C-planes, and those of the second type will be calledH-planes.

By a celebrated result of Wagner [16] any finite flag-transitive a‰ne plane will necessarily be a translation plane, and thus can be constructed from some ðn1Þ- spread of PGð2n1;qÞ. We call the spread oftypeCorHaccordingly as the asso- ciated plane is of typeCorH, respectively, as defined above. Whenn¼2, it can be shown that there are no typeC1-spreads of PGð3;qÞ, and the typeH1-spreads have been completely classified (see [3]). When n¼3, type C and type H 2-spreads of PGð5;qÞare known to exist, and the type C2-spreads of PGð5;qÞhave been clas- sified (see [1], [2], [4]). The purpose of this paper is to classify the typeH 2-spreads

* Research partly supported by NSA grant MDA 904-00-1-0029.

of PGð5;qÞ, and hence classify all odd order three-dimensional flag-transitive a‰ne planes, subject to the above gcd condition.

It should be remarked that in any dimension typeH flag-transitive a‰ne planes cannot exist for evenq, sinceqⁿþ1 must be even for such planes to exist. Moreover, typeCflag-transitive a‰ne planes are only known to exist for oddn, independent of q, although their existence for evenn>2 remains an open question.

2 Singer orbits

LetS¼PGð2n1;qÞdenote theð2n1Þ-dimensional projective space over the finite field GFðqÞ, whereqis an odd prime power andnd3is an odd integer. We modelS using the finite fieldK ¼GFðq²ⁿÞ. Letbbe a primitive element forK. Hence, the field elements

1;b;b²;. . .;b^{ðq2n1Þ=ðq1Þ1}

represent the distinct projective points ofS. The collineationy induced by multipli- cation bybis a Singer cycle ofS, and we consider the fibrationFwhose elements are the point orbits under the cyclic collineation group G generated by y^N, where N¼^q_q1ⁿ¹. Thus F has N members, each of size M¼qⁿþ1. More precisely, F¼ fWi:i¼0;1;2;. . .;N1g, where Wi¼bⁱW0 and W0¼ fb^sN :s¼0;1;2;. . .; M1g.

We start by examining how a line ofS can meet the fibrationF. The arguments are very similar to those found in [8].

Lemma 1.Let A,B,and C be three distinct collinear points ofW_tfor some t.Then the line containing A, B,and C is completely contained in W_t.Moreover, this is the only line through A which is contained inW_t.

Proof. We may assume without loss of generality that the three collinear points are inW0, and thatAis represented by the field elementb⁰¼1. ThenB andCare represented by field elements of the form b^iN and b^jN, respectively, where 1ci<

jcM1. The three points are collinear if and only if 1þab^iN ¼bb^jN, for some a;bAGFðqÞ.

Nowð1þab^iNÞ^qnþ1¼ ðbb^jNÞ^qnþ1, whereðbb^jNÞ^qnþ1¼b²b^jNM AGFðqÞ, and ð1þab^iNÞ^qnþ1 ¼ ð1þab^iNÞð1þab^iNÞ^qn

¼1þab^iNþab^iNqnþa²b^iNM: Sincea²b^iNM AGFðqÞ, we haveb^iNþb^iNqnAGFðqÞ.

Hence fðxÞ ¼ ðxb^iNÞðxb^iNqnÞis a quadratic polynomial over GFðqÞhaving b^iN as a root. Thus b^iN is in GFðq²Þ, which further implies that b^jN AGFðq²Þby the above dependency relation. Hence the three given collinear points lie on the line represented by the subfield GFðq²Þ.

To show that this line is completely contained inW₀, we observe thatðqþ1Þ jM since n is odd. Hence, as b^NM=ðqþ1Þ is a primitive element of GFðq²Þ, all points on the line represented by GFðq²Þlie in W0 by definition ofW0. The fact that there is only one line through Acompletely in W0 follows from the uniqueness of the sub-

field GFðq²Þ. r

It should be noted that the ‘‘ruling lines’’ completely contained in theG-orbitsWt

form a geometric 1-spread of PGð2n1;qÞwhenevernis odd, and there are no such ruling lines whennis even.

We now definelsto be the line joining the pointsb⁰andb^sNofW0. From the proof of the previous lemma,lsis completely contained inW0if and only ifsis a multiple of_qþ1^M .

Theorem 2.Suppose s is not a multiple of_qþ1^M ,so that l_sis not contained inW₀. 1. If s is odd,then lsis secant to exactly^qþ1₂ elements ofFand is disjoint from all other

members ofF.

2. If s is even,then lsis tangent to exactly2elements ofFand is secant to^q1₂ elements ofF.Moreover,the points of tangency correspond tob^sNGb^sNM=2.

Proof. Since s is not a multiple of _qþ1^M , we know from Lemma 1 that ls meets W0

in only two points. The argument now follows exactly as in the proof of Theorem 4

in [8]. r

Hence, from Theorem 2, we know that there are four possible ways in which a line can meet F, as listed below. We have shown that lines of the first three types exist.

Direct counting shows that lines of the fourth type also must exist for any odd integer nd3.

Ruling lines—lines completely contained in an element ofF. In particular, there are q⁴þq²þ1 such lines whenn¼3.

Secant Lines—lines which meet exactly ^qþ1₂ elements of F in two distinct points and are disjoint from the remaining N^qþ1₂ elements ofF. There are¹₂qðq1Þ ðq²þqþ1Þðq³þ1Þsuch lines whenn¼3.

Tangent Type Lines—lines which meet exactly 2 elements of Fin one point, meet exactly ^q1₂ elements of F in two distinct points, and are disjoint from the remaining N^qþ3₂ elements of F. There are ¹₂qðqþ1Þðq²þqþ1Þðq³þ1Þ such lines whenn¼3.

Purely Tangent Lines—lines which meet exactly qþ1 elements of F in a unique point and are disjoint from the remaining N ðqþ1Þelements of F. There are qðq⁴þq²þ1Þðq³þ1Þsuch lines whenn¼3.

3 The fibration in PG(5,q)

We now apply the theory from the last section to the special case when n¼3; that is,S¼PGð5;qÞ. Hence our fibration Fhas exactly N¼q²þqþ1 elements, each

containing M¼q³þ1 distinct points. As discussed in [7], there is a special collec- tion of q²þqþ1 elliptic quadrics ofS, each of which is partitioned intoqþ1 dis- tinct elements ofF. Moreover, each memberW_iofFis contained inqþ1 of these quadrics, any two of which intersect precisely inW_i. In short, if we call the elements of F‘‘points’’ and the associated quadrics ‘‘lines’’, the induced incidence structure is a projective plane of orderq.

We now defineHto be the unique Singer subgroup of index two inG. That is,His generated by the collineationy^2N. This is the subgroup associated with typeHflag- transitive a‰ne planes, as discussed in Section 1. To that end, let pbe a projective plane ofSwhich meets eachH-orbit in at most one point. Thenpmeets each of the G-orbits (i.e. elements ofF) in at most two points, as eachG-orbit is a union of two H-orbits.

Lemma 3.The planepcontains no tangent type lines.

Proof.Suppose the planepcontained a tangent type line, which we may assume isl_s for somes. From Theorem 2 we know thatsis even. But then the field elementsb⁰ andb^sN induce distinct points which are in the same orbit under the groupH, con-

tradicting the definition ofp. r

We have thus determined that the planepcan only contain secant lines and purely tangent lines, as defined at the end of the previous section.

Theorem 4.The planepis tangent to each of the q²þqþ1elements ofF.

Proof.For contradiction, suppose thatpmeets an element ofF, sayWj, in two points PandQ. Then, by Lemma 3, the linelcontainingPandQis a secant line. Now, ifp contained another secant line, saym, thenlandmmust meet in a common point, say R, contained in some element ofF, sayWk. It follows thatpmeetsWk in 3 distinct points, a contradiction. Hence,pcontains only one secant line, namely l, and there are exactly^qþ1₂ elements ofFmeetingpin exactly two points.

Recall thatW_j may be expressed as the intersection of two elliptic quadrics ofS, as discussed above, and any two such quadrics determine an algebraic pencil of quadrics in S. Moreover, the planep intersects this pencil in a planar pencil P of quadrics.

From the classification of planar pencils of quadrics [11], we know that one of the quadrics inPis a pair of distinct lines. From the discussion at the beginning of this section, this pair of lines covering 2qþ1 points meets exactly qþ1 elements ofF. Hence the planepmust meet at leastqelements ofFin two distinct points, contra- dicting our computation in the previous paragraph. That is,pcannot meet any ele- ment ofFin two distinct points, and the result follows. r

4 TypeHspreads in PG(5,q)

Let S be a type H 2-spread in S¼PGð5;qÞ, whereq is an odd prime power. In particular,Sconsists ofq³þ1 mutually disjoint planes, necessarily partitioning the

points of S. Since Singer subgroups are unique up to conjugation, we may use the Singer subgroupsH andGof the previous section. By definition of type Hspreads and the fact that the only short orbit on planes of the full Singer group is a regular 2- spread (see [10], for instance), we know thatS¼p₁^HUp₂^H, wherep₁ andp₂ are two skew planes ofS. Moreover,Sis not a single plane orbit underG. Since the planes inSare mutually disjoint,p₁(and similarly, p₂) meets each point orbit underHin at most one point. This implies by Theorem 4 thatp1(and similarly, p2) necessarily meets each of theq²þqþ1 point orbits underGin exactly one point, and thus is a

‘‘purely tangent plane’’ with respect to the fibrationFofG-orbits.

Since p1 is a purely tangent plane with respect to F, the classification of type C spreads in [2] and [4] implies that p1 is projectively equivalent (in fact, by a Singer shift) to a plane represented by a GFðqÞ-subspace of the formSb¼ fzþbz^q: zAGFðq³Þg, for some bAK¼GFðq⁶Þsuch that b^q3¼ b. Thus the fundamental issue is determining how many compatible half-spreads p₂^H exist for a given half- spreadp₁^H.

Theorem 5. Let p^H be some cyclic half-spread in PGð5;qÞ, where q is an odd prime power.Then,unlessp^H is contained in a regular spread,there is precisely one way of completingp^Hto a typeHspread and precisely one way of completingp^Hto a typeC spread.Ifp^H is contained in a regular spread,then there is only one way of completing p^H to either a typeH or type C spread, and that completion is the unique regular spread containingp^H.

Proof.As shown above,pis necessarily a purely tangent plane with respect to the fi- brationFofG-orbits, and thusp^Gis a typeCspread completing the half-spreadp^H. Moreover, we know the planepis a Singer shift of one represented by some subspace S_b, whereb^q3¼ b. Thus from the work in [12] and [14] we know there is at least one way of completingp^Hto a spread of typeH, as long asb00.

Suppose now that p₁^H and p₂^H are two completions of the half-spread p^H to a spread that is either of typeHor of typeC. Then by the same argument as that given above, we know that p1 andp2 must be purely tangent planes with respect to the fibration F. By translating appropriately and shifting our point of view, we may assume bothp1 andp2 pass through the point represented by b⁰, and hence each is represented by a subspace of the form_1þb¹ S_b¼n^xþbx_1þb^q:xAGFðq³Þo

, for somebAK withb^q3¼ b. Note that the collineation induced by multiplication byð1þbÞ¹ is a power of the Singer cycle, and the shifted plane ð1þbÞ¹Sb is a purely tangent plane passing throughb⁰. With a bit of computation, which will soon become ap- parent, one can show that distinct values ofbdetermine distinct planesð1þbÞ¹S_b, and hence we get as a byproduct that there are exactly q³ purely tangent planes through a given point, such asb⁰, for any odd prime powerq.

Let B¼ fb:bAK;b^q3¼ bg. Now the existence of the above two completions of p^H implies the existence of b1;b2AB such that ð1þb1Þ¹Sb1 and ð1þb2Þ¹Sb2

represent planes, namelyp1 andp2, that meet the same point orbits underH. Since H ¼hy^{2ðq2þqþ1Þ}i, this means that for eachxAGFðq³Þ, there exists some yAGFðq³Þ such that

xþb1x^q 1þb₁ q³þ1

¼g yþb2y^q 1þb₂ q³þ1

for some nonzero squaregin GFðqÞ. Moreover, we may chose an appropriate non- zero GFðqÞ-multiple of yso thatg¼1. Thus, for everyxAGFðq³Þ, there exists some

yin GFðq³Þsuch that

x²b₁²x^2q

1b₁² ¼y²b₂²y^2q

1b²₂ ; ð1Þ

whereb1andb2are fixed elements ofB.

For each bAB, consider the mappingTb:GFðq³Þ !GFðq³ÞviaTb:z!^zb_1b^2z2^q. Sinceb² is either 0 or a nonsquare in GFðq³Þ,Tbis a nonsingular linear transforma- tion of the vector space GFðq³Þover GFðqÞ, and henceTb is a permutation polyno- mial of the field GFðq³Þ. Moreover, eachT_bfixes each of the elements 0 and 1. Hence the composition T_b1T_b¹

2 is a GFðqÞ-linear permutation polynomial with the pre- viously mentioned properties that also maps squares to squares by Equation (1), and thus preserves quadratic character. By the main theorem in Carlitz [6] T_b1T_b¹

2 is necessarily one of the following three maps:z!z,z!z^q, orz!z^q2. In particular, this implies thatx²¼y²,x² ¼y^2q, orx²¼y^2q2in Equation (1) above. Now take an appropriate linear combination to solve forx², say ¹

ð1b₁²Þ^q2þqtimes Equation (1) plus

b₁²

ð1b²₁Þ^q2þ1 times the q-th power of (1) plus ^b

2ðqþ1Þ 1

ð1b₁²Þ^qþ1 times the q²-th power of (1), and simplify. One obtains the equation

1b₁^{2ðq2þqþ1Þ}

ð1b₁²Þ^q2þqþ1x²¼ 1b₁^{2ðq2þqþ1Þ}

ð1b₁^2qÞð1b₁^2q2Þð1b₂²Þþb^2ðqþ1Þ₁ N

! y² þN^qy^2qþb²₁N^q2y^2q2;

where N¼ ^b

2q2 1

ð1b₂²Þð1b₁^2qÞð1b^2q₁²Þ ^b

2q2 2

ð1b^2q₂²Þð1b^2q₁Þð1b₁²Þ. By the Carlitz result above, we must haveN¼0 and thus_1b^x22

1

¼_1b^y22 2

. Since Tb1T_b¹₂ fixes the element 1, we must have x²¼1 if and only if y²¼1, and thusb₁¼Gb₂.

This implies that there are precisely two ways of completing the half-spread p^H to a full spread of type Cor type H, provided b₁00. Namely, if we think of p₁ as p^yq

2þqþ1

, then choosing b2¼b1 will yield the type C completion p^HUp₁^H, while choosingb2¼ b1will yield the typeHcompletionp^HUp₂^Has described in [12] and [14]. Ifb1 ¼0, then necessarilyb2¼0 and the only possible completion is the regular

spread (see [12]). This completes the proof. r

Corollary 6.All odd order three-dimensional flag-transitive a‰ne planes of typeHor Care known.If the order is q³,so that the kernel isGFðqÞ,the number of isomorphism

classes of each type is at least ðq1Þ=2e, where q¼p^eand p is an odd prime.Fur- thermore,if q is a prime,then the number of isomorphism classes of each type is exactly ðq1Þ=2.Finally,ifgcd¹₂ðq³þ1Þ;3e

¼1,then with the exception of Hering’s plane of order27every non-Desarguesian flag-transitive a‰ne plane of order q³ with kernel GFðqÞis necessarily of typeCor typeH,and hence is known.

Proof.The classification of typeCflag-transitive a‰ne planes of odd orderq³follows from the work in [1], [2], and [4] as discussed above, while the classification in the type H case follows from the previous theorem. The isomorphism counts are given in [12] and [14] (see also [2]). The final statement is a special case of Lemma 1

in [9]. r

Thus, modulo the gcd condition stated in the above corollary, the only possible flag-transitive a‰ne planes of odd order q³ are those constructed by Kantor and Suetake in [12] and [14], together with Hering’s plane of order 27. When q¼5, the gcd condition is not satisfied, and yet exhaustive searching in [15] shows that there are no other flag-transitive a‰ne planes of order 125. Perhaps the gcd condition is superfluous.

5 Concluding remarks

The key to classifying the typeHflag-transitive a‰ne planes of odd orderq³is show- ing that ifp^His a half-spread, thenpmust be tangent to each of theG-orbits in the fibrationF. Once this is known, a short proof of the classification in both the typeC and typeHcases could be given if one could prove directly that there are preciselyq³ purely tangent planes with respect toFthat pass through a given point, sayb⁰. This would eliminate the reduction to Baer subplane partitions in [1], the very involved use of linearized polynomials in [2], and the messy cyclotomic lemma in [4]. Unfortu- nately, we have so far been unable to do this counting directly.

Whenqis even, computer searches using MAGMA [5] show that for small values ofqthere are preciselyðq²þ1Þðqþ1Þpurely tangent planes with respect to the fibra- tion F that pass through a given point, say b⁰. Again, if one could directly prove that this result always holds, one would essentially have a classification of the flag- transitive a‰ne planes of type C with even order q³ (there are none of type H).

Namely, based on the known construction (succinctly described in [13]), one can show that there are at leastðq²þ1Þðqþ1Þpurely tangent planes passing throughb⁰ whenqis even, and hence all would be known.

In higher dimensions the problem is trickier. Consider a type H flag-transitive a‰ne plane of order qⁿ, where q is necessarily odd, and letp^H be one of the asso- ciated cyclic half-spreads of PGð2n1;qÞ. Ifnis odd, in all known examplespmeets each G-orbit in exactly one point and thus is ‘‘purely tangent’’ with respect to the fibrationF. Hence there is a companion typeCflag-transitive a‰ne plane with asso- ciated spreadp^G. Whenn¼3, we used (in Section 3) the classification of pencils of

quadrics in PGð2;qÞto show that such an intersection pattern must happen. Proving that this pattern must hold for all odd n>3 would be a big step toward a higher- dimensional classification. Perhaps this could be done without the complete classifi- cation of pencils of quadrics in higher dimensions, which is as yet unknown.

Whenn is even, all known examples of type H flag-transitive a‰ne planes with associated spreadp₁^HUp₂^Hare such thatp₁meets half theG-orbits ofFin two points each (from di¤erent H-orbits), while p2 meets the other half of the G-orbits in two points each. Whenn¼2,p1 andp2 are lines and it is straightforward to show that this must be the case (see [8]). For evennd4, it is unclear at present why (or if ) this pattern must hold. In particular, it would be nice to know (for q odd orq even) if there exists a purely tangentðn1Þ-space with respect to Ffor even nd4, other than the example where the resulting cyclic spread is regular and hence the plane is Desarguesian.

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Received 20 September, 2002

R. D. Baker, Department of Mathematics, West Virginia State College Institute, WV 25112- 1000, USA

Email: [email protected]

C. Culbert, G. L. Ebert, Department of Mathematical Sciences, University of Delaware, Newark, DE 19716-2553, USA

Email: [email protected]

K. E. Mellinger, Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607-7045, USA

Email: [email protected]