WilliamE.Cherowitzo,ChristineM.O’KeefeandTimPenttila* q -clansincharacteristic2 Aunifiedconstructionoffinitegeometriesassociatedwith

(de Gruyter 2003

A unified construction of finite geometries associated with q-clans in characteristic 2

William E. Cherowitzo, Christine M. O’Keefe and Tim Penttila*

(Communicated by G. Korchma´ros)

Dedicated to Louis Reynolds Antoine Casse

Abstract.Flocks of Laguerre planes, generalized quadrangles, translation planes, ovals, BLT- sets, and the deep connections between them, are at the core of a developing theory in the area of geometry over finite fields. Examples are rare in the case of characteristic two, and it is the purpose of this paper to contribute a fifth infinite family. The approach taken leads to a unified construction of this new family with three of the previously known infinite families, namely those satisfying a symmetry hypothesis concerning cyclic subgroups of PGLð2;qÞ. The calcu- lation of the automorphisms of the associated generalized quadrangles is su‰cient to show that these generalized quadrangles and the associated flocks and translation planes do not belong to any previously known family.

Key words.Flocks, ovals, herds, hyperovals, Adelaideq-clans, translation planes, generalized quadrangles.

2000 Mathematics Subject Classification. Primary 51E21, 51E20, 51E12, 51E15; Secondary 05B25

1 Introduction

The last twenty years have seen extensive activity in the area of finite geometry, with the publication of many papers dealing with the connections between flocks of Laguerre planes, generalized quadrangles, translation planes, ovals and BLT- sets. In this setting, we identify some significant papers as follows. First, the sequence of exchanges between Kantor and Payne delineated the evolution of the concept of a q-clan and the construction of generalized quadrangles as group coset geo- metries [14, 23, 33, 15, 24], while Thas [40] recognised the connection between flocks of Laguerre planes and generalized quadrangles. In the case of odd characteristic, Bader, Lunardon and Thas [2] introduced BLT-sets and Knarr [16] developed a geo-

* This work was supported by the Australian Research Council.

metric construction of the generalized quadrangle from the BLT-set. In the case of even characteristic, Payne and Cherowitzo, Penttila, Pinneri and Royle [24, 6] intro- duced herds of ovals and elucidated their connection with the generalized quadran- gles. The sequence of papers by Thas [41, 42, 43] on translation generalized quad- rangles of order ðs;s²Þ culminated in a geometric construction of the generalized quadrangle from the flock in all characteristics. Finally, the connection between ovals and flocks of translation Laguerre planes was explained by Cherowitzo [5].

In the case of odd characteristic, the BLT-set and the Knarr construction provide useful insights into the connections between the geometries mentioned above, while recent constructions of families [34, 18] as well as many sporadic examples over small fields [38, 17] suggest that there may in fact be an embarassment of riches. The situ- ation in characteristic two stands in stark contrast: there are only four known fami- lies, to which we add a fifth in this paper, and no known sporadic example. In com- pensation for this paucity of examples in characteristic two, there is a rich association with ovals and hyperovals which is missing in odd characteristic. The ovals so arising, in turn, lead to further generalized quadrangles and Laguerre planes. Furthermore, characterisation theorems have been proved in characteristic two [13, 9, 19, 39, 20], suggesting that the rarity of examples in characteristic two is not merely temporary.

Despite the recent activity and advances, there is much work still to be done, even in characteristic two. In this case, O’Keefe and Penttila [20] obtained charac- terisations with symmetry hypotheses concerning subgroups of order qandq1 of PGLð2;qÞ, leaving the (cyclic) subgroups of order qþ1 still to be dealt with. The cyclic hypothesis was first suggested by the Subiaco examples [6, 27, 1, 30] and the examples for q¼4³ and 4⁴ discovered by Penttila and Royle [37]. The di¤erent approach to automorphism calculations possible in characteristic two [28] led to a further exploration by Payne, Penttila and Royle [31] and to further examples for q¼4⁵;4⁶;4⁷ and 4⁸. In this paper we generalise these six examples to a new infinite family. We remark that the cyclic hypothesis is satisfied by three of the previously known families (the exception is the family of Payne [24]), and this paper contributes a unified construction of these three families with the new family. This work should contribute to an eventual classification in this case, characterising these four families by the cyclic hypothesis.

Finally, another persistent thread in the recent literature concerns the automor- phisms of the associated geometries investigated by Payne and several others [32, 26, 27, 1, 30, 28, 29, 22]. The recent results by O’Keefe and Penttila [20] allow the inter- pretation of the cyclic hypothesis in the herd model. This interpretation is important because it led to the unified construction contained in this paper, as was suggested at the end of the introduction of [20]. In the current context, we also use O’Keefe and Penttila’s techniques [20] to calculate the groups of the geometries we construct, and hence to show that the associated generalized quadrangles, flocks and transla- tion planes are new. We do not yet have a proof that the associated hyperovals do not belong to the previously known families, although this is true for those over the fields of orders 4³ and 4⁴. Such a proof would require more information concern- ing the groups of the hyperovals. Here it is appropriate to note that the groups of the Cherowitzo hyperovals [4] have yet to be determined, although the partial results of

O’Keefe and Thas [21] were enough for Penttila and Pinneri [35] to show that the Cherowitzo hyperovals were new.

2 Preliminaries

Let C¼ fAt:tAGFðqÞgbe a collection of 22 matrices with entries from GFðqÞ.

Following Payne [25], we call Ca q-clanifAsAt is anisotropic (that is, the equa- tion ðx;yÞðAsAtÞðx;yÞ^T ¼0 has only the trivial solution ðx;yÞ ¼ ð0;0Þ) for all s;tAGFðqÞwiths0t.

We will use the (absolute) trace function on GFðqÞ, q¼2^e, as follows. Let trace:GFðqÞ !GFðqÞbe defined by traceðxÞ ¼xþx²þ þx^2e1.

As is discussed by Cherowitzo, Penttila, Pinneri and Royle [6, 2.1], without loss of generality we can assume that

At¼ fðtÞ t¹⁼² 0 agðtÞ

whereaAGFðqÞsatisfies traceðaÞ ¼1,f;g:GFðqÞ !GFðqÞare functions satisfying fð0Þ ¼gð0Þ ¼0 and fð1Þ ¼gð1Þ ¼1 and

trace aðfðsÞ þfðtÞÞðgðsÞ þgðtÞÞ sþt

¼1 ð2:1Þ

for alls;tAGFðqÞwiths0t.

Conversely, if there exist functions f;g:GFðqÞ !GFðqÞ with fð0Þ ¼gð0Þ ¼0 and fð1Þ ¼gð1Þ ¼1 and an elementaAGFðqÞwith traceðaÞ ¼1 such that Equation (2.1) holds, then the set of matrices fðtÞ t¹⁼²

0 agðtÞ

:tAGFðqÞ

is aq-clan.

We call a q-clan normalised if it is written in this standard form and note that it follows immediately from Equation (2.1) that f andgare permutation polynomials.

In the following subsections we show howq-clans,qeven, can be used to construct various important geometric structures, thus motivating their study. We then survey the knownq-clans,qeven, to the time of preparation of this paper.

2.1 Flocks of quadratic cones. Let O be an oval in PGð2;qÞ, and let PGð2;qÞ be embedded as a hyperplane in PGð3;qÞ. For a pointvAPGð3;qÞnPGð2;qÞ, the union of the points on the lines incident withvand a point of Ois the conewithvertex v andbaseO. Aquadratic coneis a cone with baseOa (non-degenerate) conic. Aflock of a cone Kwith vertex vis a set ofq planes which partitionsKnfvg into disjoint ovals. IfLis a line of PGð3;qÞhaving no point in common withKthen theqplanes throughLand notvform a flock. Such a flock is calledlinear, and forq¼2;3 and 4 every flock of a cone is linear [40].

Theorem 2.1 ([24, 40]). Let q be even and let K be the quadratic cone in PGð3;qÞ with equation X₀X₁¼X₂²; thus the vertex is v¼ ð0;0;0;1Þ. The set of planes F¼

fatX₀þc_tX₁þb_tX₂þX₃¼0:tAGFðqÞg is a flock of K if and only if b_t0b_s for s0t and

trace ðasþa_tÞðcsþc_tÞ ðbsþb_tÞ²

!

¼1 for all s0t:

By Equation (2.1), withqeven, we have:C¼ a_t b_t 0 c_t

:tAGFðqÞ

is aq-clan if and only if F¼ fatX₀þc_tX₁þb_tX₂þX₃¼0:tAGFðqÞg is a flock of the qua- dratic coneKin PGð3;qÞwith equationX0X1¼X₂².

2.2 Elation generalized quadrangles. Let G¼ fða;c;bÞ:a;bAGFðqÞ²;cAGFðqÞg, with multiplication defined as:

ða;c;bÞða⁰;c⁰;b⁰Þ ¼ ðaþa⁰;cþc⁰þba⁰;bþb⁰Þ;

where (sincea;bare 2-tuples of elements of GFðqÞ) we define:

ba¼ ffiffiffiffiffiffiffiffiffiffiffiffi bPa^T

p ; withP¼ 0 1 1 0

:

LetC¼ fAt:tAGFðqÞgbe a normalisedq-clan, and define the following subgroups ofG:

AðyÞ ¼ fð0;0;bÞ:bAGFðqÞ²g and AðtÞ ¼ fða; ffiffiffiffiffiffiffiffiffiffiffiffiffi

aA_ta^T

p ;t¹⁼²aÞ:aAGFðqÞ²g; tAGFðqÞ:

For eachtAGFðqÞUfygwe defineAðtÞ ¼AðtÞZwhereZ¼ fð0;c;0Þ:cAGFðqÞg is the centre ofG. ThenF¼ fAðtÞ:tAGFðqÞUfyggis a 4-gonal familyforG[24], see [33, 10.4].

Starting with this 4-gonal family, Kantor’s [14] construction gives an elation generalized quadrangle of orderðq²;qÞ(with base pointðyÞ) on whichGacts by left multiplication as a group of elations, see [33, 8.2], as follows:

points: (i) elements gAG, (ii) cosets gAðtÞ forgAG, tAGFðqÞUfyg and (iii) a symbolðyÞ,

lines: (a) cosets gAðtÞ for gAG, tAGFðqÞUfyg and (b) symbols ½AðtÞ for tA GFðqÞUfyg.

(Here we use left cosets, in contrast to some of the literature which uses right cosets, because throughout this papergroups are acting on the left.) A point gof type (i) is incident with each linegAðtÞfortAGFðqÞUfyg. A pointgAðtÞof type (ii) is inci- dent with the line½AðtÞand with each linehAðtÞcontained ingAðtÞ. The pointðyÞ is incident with each line½AðtÞfortAGFðqÞUfyg. There are no further incidences.

We have therefore outlined a process by which a (normalised)q-clanCgives rise to a 4-gonal family for the group Gabove, and hence to an elation generalized quad- rangle GQðCÞof orderðq²;qÞ.

2.3 Herds of ovals.Aherd[6] of ovals in PGð2;qÞ,qeven, is a family ofqþ1 ovals fOs:sAGFðqÞUfygg, each of which has nucleus ð0;0;1Þ, contains the points ð1;0;0Þ;ð0;1;0Þandð1;1;1Þ, and is such that

Oy ¼ fð1;t;gðtÞÞ:tAGFðqÞgUfð0;1;0Þg and Os¼ fð1;t;fsðtÞÞ:tAGFðqÞgUfð0;1;0Þg;

where

fsðtÞ ¼f₀ðtÞ þasgðtÞ þs¹⁼²t¹⁼² 1þasþs¹⁼² for someaAGFðqÞsatisfying traceðaÞ ¼1.

Theorem 2.2 ([24, 6]). Let q be even. Let f0;g:GFðqÞ !GFðqÞ be functions with f0ð0Þ ¼gð0Þ ¼0and f0ð1Þ ¼gð1Þ ¼1.There exists aAGFðqÞwithtraceðaÞ ¼1and such that

trace aðf0ðsÞ þf0ðtÞÞðgðsÞ þgðtÞÞ ðsþtÞ

¼1 for all s0t

if and only iffOs:sAGFðqÞUfyggis a herd,where Oy¼ fð1;t;gðtÞÞ:tAGFðqÞgUfð0;1;0Þg and

Os¼ 1;t;f₀ðtÞ þasgðtÞ þs¹⁼²t¹⁼² 1þasþs¹⁼²

:tAGFðqÞ

Ufð0;1;0Þg:

By Equation (2.1), withq even, we have:C¼ f₀ðtÞ t¹⁼² 0 agðtÞ

:tAGFðqÞ

is a q-clan if and only iffOs:sAGFðqÞUfyggis a herd.

2.4 Translation planes. Thas [8] and Walker [44] independently showed how to construct a translation plane from a flock of a quadratic cone. We now give a brief outline of this construction, and note that relevant details and background can be found in [10, 12].

Let FðCÞ be the flock of the quadratic cone K of PGð3;qÞ with equation X₀X₁¼X₂², associated with theq-clanC. EmbedKinto the Klein Quadric Q^þð5;qÞ

in PGð5;qÞand let ?denote the polarity of PGð5;qÞassociated with Q^þð5;qÞ. The set of pointsW¼6

pAFðCÞp^?VQ^þð5;qÞis an ovoid of Q^þð5;qÞ, and we letSbe the spread of PGð3;qÞassociated with Wvia the Klein correspondence. By the Andre`–

Bruck Bose correspondence, fromSthere arises a translation planepðCÞof orderq² with kernel containing GFðqÞ.

2.5 The knownq-clans,qeven.In this section we list the knownq-clans withq¼2^e, up to isomorphism of the associated generalized quadrangle.

Theclassical q-clan associated with a linear flock [40], for allq¼2^e, is

C¼ t¹⁼² t¹⁼² 0 at¹⁼²

:tAGFðqÞ

foraAGFðqÞwith traceðaÞ ¼1. The associated GQ is isomorphic to the GQ com- prising the points and lines of the Hermitian variety Hð3;q²Þ(see [33, 3.1.1]) and the associated translation plane is Desarguesian. The associated herd isfOs:sAGFðqÞU fygg, whereOs¼ fð1;t;t¹⁼²Þ:tAGFðqÞgUfð0;1;0Þgfor allsAGFðqÞUfyg.

TheFTWKB q-clan, forq¼2^ewitheodd, is

C¼ t¹⁼⁴ t¹⁼² 0 t³⁼⁴

:tAGFðqÞ

and is classical if and only if q¼2. The associated flock arises by the geometrical construction of Fisher and Thas [8, Theorem 3.10]; in this case the corresponding translation plane was discovered by Walker [44] (using flocks) and independently by Betten [3]. The GQ was discovered by Kantor [14] and the herd comprisesqþ1 translation ovals, each projectively equivalent tofð1;t;t¹⁼⁴Þ:tAGFðqÞgUfð0;1;0Þg.

ThePayne q-clan [24], forq¼2^ewitheodd, is

C¼ t¹⁼⁶ t¹⁼² 0 t⁵⁼⁶

:tAGFðqÞ

:

It is classical if and only ifq¼2, and FTWKB if and only if q¼8. The herd com- prises two Segre–Bartocci ovals (equivalent to fð1;t;t¹⁼⁶Þ:tAGFðqÞgUfð0;1;0Þg) andq1 further ovals now known as Payne ovals. In this case there is (up to iso- morphism) one associated GQ and translation plane, but two associated flocks, see [26].

TheSubiaco q-clan [6], forq¼2^e, is C¼Cd¼ f0ðtÞ t¹⁼²

0 agðtÞ

:tAGFðqÞ

;

where, for somedAGFðqÞwithd²þdþ100 and traceð1=dÞ ¼1, we have

a¼d²þd⁵þd¹⁼² dð1þdþd²Þ

f0ðtÞ ¼d²ðt⁴þtÞ þd²ð1þdþd²Þðt³þt²Þ

ðt²þdtþ1Þ² þt¹⁼² and gðtÞ ¼d⁴t⁴þd³ð1þd²þd⁴Þt³þd³ð1þd²Þt

ðd²þd⁵þd¹⁼²Þðt²þdtþ1Þ² þ d¹⁼²

ðd²þd⁵þd¹⁼²Þt¹⁼²:

It is classical if and only if q¼2 and is FTWKB (or Payne) if and only if q¼8.

There is (up to isomorphism) one associated flock and GQ, and ife12ðmod 4Þthen there are two associated ovals, otherwise only one. The flock in the caseq¼16 is due to De Clerck and Herssens [7], and the associated herd comprises 17 Lunelli–Sce ovals. In the general case, the associated herd comprises qþ1 ovals which are now known as Subiaco ovals.

Finally, there are further examples for q¼4³;4⁴;4⁵;4⁶;4⁷ and 4⁸ discovered in the series of three papers [36, 37, 31]. The main result of this paper is the construc- tion of an infinite family ofq-clans which includes these examples. Our construction in fact gives a unified presentation of the classical, FTWKB, Subiaco and the new infinite family ofq-clans.

3 The Adelaideq-clans

The purpose of this section is to prove our main theorem, Theorem 3.1. After the statement, we will proceed via a sequence of lemmas.

Theorem 3.1. Let GFðq²Þ be a quadratic extension of GFðqÞ with q¼2^e. Let bAGFðq²Þnf1gbe such thatb^qþ1 ¼1,and let TðxÞ ¼xþx^q for all xAGFðq²Þ.Let aAGFðqÞand the functions f;g:GFðqÞ !GFðqÞbe defined by:

a¼Tðb^mÞ TðbÞ þ 1

Tðb^mÞþ1 fðtÞ ¼ f_m;bðtÞ ¼Tðb^mÞðtþ1Þ

TðbÞ þ Tððbtþb^qÞ^mÞ

TðbÞðtþTðbÞt¹⁼²þ1Þ^m1þt¹⁼² and

agðtÞ ¼ag_m;bðtÞ ¼Tðb^mÞ

TðbÞ tþ Tððb²tþ1Þ^mÞ

TðbÞTðb^mÞðtþTðbÞt¹⁼²þ1Þ^m1þ 1 Tðb^mÞt¹⁼² and let

C¼Cm;b ¼ fðtÞ t¹⁼² 0 agðtÞ

:tAGFðqÞ

:

If m1G1ðmodqþ1Þ then C is the classical q-clan for all q¼2^e and for all b. If q¼2^e with e odd and m1G^q

2 ðmodqþ1Þ then C is the FTWKB q-clan for all

b.If q¼2^ewith e>2and m1G5ðmodqþ1ÞthenCis the Subiaco q-clan for allb such that ifl is a primitive element ofGFðq²Þ,so that b¼l^kðq1Þ,then qþ1akm.

If q¼2^ewith e>2even and m1G^q1

3 ðmodqþ1ÞthenCis a q-clan,which we call the Adelaide q-clan,for allb.

First we introduce some notation which will be used throughout this section.

LetK¼GFðq²Þbe an extension field ofF¼GFðqÞwithq¼2^e. We let the relative trace function fromKtoF be denoted byT, that is,T:K!F is defined byTðxÞ ¼ xþx^qand let the absolute trace function onF be denoted by traceF.

LetbAKnf1gbe an element of norm 1 relative toF, that is,b^qþ1¼1. We define three auxiliary functionsh₁;h₂;h₃:F!F by

h₁ðtÞ ¼ Tððbtþb^qÞ^mÞ ðtþTðbÞt¹⁼²þ1Þ^m1 h₂ðtÞ ¼ Tððb²tþ1Þ^mÞ

ðtþTðbÞt¹⁼²þ1Þ^m1 and

h₃ðtÞ ¼Tðb^mÞh1ðtÞ þh₂ðtÞ ¼ Tððtþb²Þ^mÞ

ðtþTðbÞt¹⁼²þ1Þ^m1; ð3:1Þ where mAf1;. . .;q²2g. Notice that tþTðbÞt¹⁼²þ1 is non-zero for all tAF.

To see this, since the roots of tþTðbÞt¹⁼²þ1 are b² and b^2q, it su‰ces to show thatbBF. Letlbe a primitive element ofK, sob¼l^kðq1Þfor somekAf1;. . .;qg.

ThenbAFif and only ifb^q1¼1, which is if and only ifqþ1jkðq1Þ². But this is impossible sinceðqþ1;q1Þ ¼1 andkAf1;. . .;qg.

Finally, we defineaAGFðqÞand the functions f;g:F !F by:

a¼Tðb^mÞ TðbÞ þ 1

Tðb^mÞþ1 fðtÞ ¼ f_m;bðtÞ ¼Tðb^mÞ

TðbÞ ðtþ1Þ þ 1

TðbÞh₁ðtÞ þt¹⁼² and

agðtÞ ¼agm;bðtÞ ¼Tðb^mÞ

TðbÞ tþ 1

TðbÞTðb^mÞh2ðtÞ þ 1 Tðb^mÞt¹⁼²: Lemma 3.2.With the notation as above,and for all t;sAF with t0s we have

trace_F ðfðtÞ þfðsÞÞðagðtÞ þagðsÞÞ tþs

¼traceF 1þ 1

Tðb^mÞþTððMtM_s^qÞ^mÞ

TðMtMs^qÞ þTðM_t^mÞ

TðMtÞ þTðM_s^mÞ TðMsÞ

;

where

M_t¼ t¹⁼²þb t¹⁼²þb^q: Proof.First, write

ðfðtÞ þfðsÞÞðagðtÞ þagðsÞÞ

tþs ¼ 1

tþsAB;

where

A¼Tðb^mÞðtþsÞ þh₁ðtÞ þh₁ðsÞ

TðbÞ þ ðtþsÞ¹⁼² and B¼Tðb^mÞðtþsÞ þ_Tðb¹mÞðh2ðtÞ þh2ðsÞÞ

TðbÞ þ 1

Tðb^mÞðtþsÞ¹⁼²:

As we calculate the absolute trace of this expression we shall encounter terms of absolute trace 0 (with respect toF). Once such terms have been identified they will be accumulated in a single term denoted by C, thus C is not constant throughout the calculation, but at all times trace_FðCÞ ¼0. On expanding the product we obtain

Tðb^mÞ²

TðbÞ² ðtþsÞ þTðb^mÞ

TðbÞ ðtþsÞ¹⁼²þh2ðtÞ þh2ðsÞ

TðbÞ² þTðb^mÞðh1ðtÞ þh1ðsÞÞ TðbÞ² þðh1ðtÞ þh₁ðsÞÞðh2ðtÞ þh₂ðsÞÞ

TðbÞ²Tðb^mÞðtþsÞ þh₁ðtÞ þh₂ðtÞ þh₁ðsÞ þh₂ðsÞ TðbÞTðb^mÞðtþsÞ¹⁼² þ 1

TðbÞðtþsÞ¹⁼²þ 1

Tðb^mÞ: ð3:2Þ

The sum of the first two terms is an element of absolute trace 0, and is thus incorporated into C. By (3.1) the sum of the third and fourth terms is

1

TðbÞ²ðh3ðtÞ þh₃ðsÞÞ. Now,

h₁ðtÞh2ðtÞ ¼ Tððbtþb^qÞ^mÞ

ðtþTðbÞt¹⁼²þ1Þ^m1 Tððb²tþ1Þ^mÞ ðtþTðbÞt¹⁼²þ1Þ^m1

¼Tðb^mÞðtþTðbÞt¹⁼²þ1Þ^2mþb^mðb^2qtþ1Þ^2mþb^qmðb²tþ1Þ^2m ðtþTðbÞt¹⁼²þ1Þ^2m2

¼Tðb^mÞðt²þTðbÞ²tþ1Þ þh₁²ðtÞ þh₂²ðtÞ Tðb^mÞ :

Hence, the fifth term of (3.2) can be written as:

ðh1ðtÞ þh1ðsÞÞðh2ðtÞ þh2ðsÞÞ TðbÞ²Tðb^mÞðtþsÞ ¼

h₁²ðtÞþh₂²ðtÞþh₁²ðsÞþh₂²ðsÞ Tðb^mÞ

TðbÞ²Tðb^mÞðtþsÞ

þTðb^mÞððtþsÞ²þTðbÞ²ðtþsÞÞ þh1ðtÞh2ðsÞ þh2ðtÞh1ðsÞ TðbÞ²Tðb^mÞðtþsÞ

¼h₁²ðtÞ þh₂²ðtÞ þh₁²ðsÞ þh₂²ðsÞ

TðbÞ²Tðb^mÞ²ðtþsÞ þ tþs

TðbÞ²þ1þh1ðtÞh2ðsÞ þh2ðtÞh1ðsÞ TðbÞ²Tðb^mÞðtþsÞ : We see that the first term of this expression is the square of the sixth term in (3.2), and the second term here is the square of the seventh term in (3.2). Thus, there are two more expressions of absolute trace 0 that can be incorporated intoC. Rewriting (3.2), we now obtain

1þ 1

Tðb^mÞþh₃ðtÞ þh₃ðsÞ

TðbÞ² þh₁ðtÞh2ðsÞ þh₂ðtÞh1ðsÞ

TðbÞ²Tðb^mÞðtþsÞ þC: ð3:3Þ Let

M_t ¼ tþb²

tþTðbÞt¹⁼²þ1¼ ðt¹⁼²þbÞ²

ðt¹⁼²þbÞðt¹⁼²þb^qÞ¼ t¹⁼²þb t¹⁼²þb^q: It follows that

h3ðtÞ

TðbÞ²¼ Tððtþb²Þ^mÞ TðbÞ²ðtþTðbÞt¹⁼²þ1Þ^m1

¼ðtþTðbÞt¹⁼²þ1ÞðM_t^mþM_t^qmÞ

TðbÞ² :

Then, because

MtþM_t^q¼ t¹⁼²þb

t¹⁼²þb^qþt¹⁼²þb^q t¹⁼²þb

¼tþb²þtþb^2q

tþTðbÞt¹⁼²þ1¼ TðbÞ² tþTðbÞt¹⁼²þ1; we have

h₃ðtÞ

TðbÞ²¼TðM_t^mÞ TðMtÞ :

A simple calculation shows that:

h₁ðtÞh2ðsÞ þh₂ðtÞh1ðsÞ

¼Tðb^mÞTðððtþb²Þðsþb^2qÞÞ^mþ ððtþb^2qÞðsþb²ÞÞ^mÞ ððtþTðbÞt¹⁼²þ1Þ^m1ðsþTðbÞs¹⁼²þ1ÞÞ^m1

¼Tðb^mÞðtþTðbÞt¹⁼²þ1ÞðsþTðbÞs¹⁼²þ1ÞTððMtM_s^qÞ^mÞ: Then, because

TðMtM_s^qÞ ¼ ðt¹⁼²þbÞ ðt¹⁼²þb^qÞ

ðs¹⁼²þb^qÞ

ðs¹⁼²þbÞ þðt¹⁼²þb^qÞ ðt¹⁼²þbÞ

ðs¹⁼²þbÞ ðs¹⁼²þb^qÞ

¼ðtþb²Þðsþb^2qÞ þ ðtþb^2qÞðsþb^qÞ ðtþTðbÞt¹⁼²þ1ÞðsþTðbÞs¹⁼²þ1Þ

¼ TðbÞ²ðtþsÞ

ðtþTðbÞt¹⁼²þ1ÞðsþTðbÞs¹⁼²þ1Þ; we have

h₁ðtÞh2ðsÞ þh₂ðtÞh1ðsÞ

TðbÞ²Tðb^mÞðtþsÞ ¼TððMtM_s^qÞ^mÞ TðMtMs^qÞ : We can now rewrite (3.3) as:

1þ 1

Tðb^mÞþTððMtM_s^qÞ^mÞ

TðMtMs^qÞ þTðM_t^mÞ

TðMtÞ þTðM_s^mÞ TðMsÞ þC:

and applying the absolute trace function trace_F gives the desired result. r Let N₁¼ fgAKnf1g jg^qþ1¼1g. Observe that for anytAF andbAN₁ we have M_tAN₁, and that fort;sAF andbAN₁ we haveM_tM_s^qAN₁ providedt0s.

Lemma 3.3.For q¼2^e,if there exists a constant cAf0;1gsuch that traceF

Tðg^mÞ TðgÞ

¼c; for allgAN1; then

traceF

ðfðtÞ þfðsÞÞðagðtÞ þagðsÞÞ tþs

¼traceF

1 Tðb^mÞ

holds

for all e; if c¼traceFð1Þ;

for all even e; if c¼0;

for all odd e; if c¼1:

8>

<

>:

Proof.This follows immediately from Lemma 3.2, the additivity of the trace function and the fact that traceFð1Þ ¼1 if and only ifeis odd. r Lemma 3.4. Let l be a primitive element of K so that N1¼ fl^kðq1Þ:k¼1;. . .;qg.

Then

trace_F 1 Tðb^mÞ

¼1 ð3:4Þ

for allb¼l^kðq1ÞAN1such that qþ1akm.In particular,ifðm;qþ1Þ ¼1then(3.4) holds for allbAN₁.

Proof. First note that trace_Fð1=Tðb^mÞÞ ¼1 if and only if the quadratic x²þTðb^mÞxþ1 is irreducible over F. But this quadratic has roots b^m and b^qm, so trace_Fð1=Tðb^mÞÞ ¼1 if and only if b^mBF. Now b^m¼l^kmðq1ÞAN₁ is an ele- ment of F if and only if l^kmðq1Þ2 ¼1. Hence if b^mAF then ðqþ1Þ jkm, since ðq1;qþ1Þ ¼1. We have shown that ifb¼l^kðq1ÞandmAf1;. . .;q²2gsatisfy qþ1akm then trace_Fð1=Tðb^mÞÞ ¼1, as required. Noticing that if ðm;qþ1Þ ¼1

thenqþ1akmgives the last statement. r

It is immediate from this lemma that, given K, one can always find a bAN₁ such that (3.4) holds. Although we permit the exponent m to take any value in

f1;. . .;q²2g, not all values ofmgive di¤erent functions f andg.

Lemma 3.5.For the functions f and g defined above,we have

fðtÞ ¼ fm;bðtÞ ¼ fmþkðqþ1Þ;bðtÞ and gðtÞ ¼gm;bðtÞ ¼gmþkðqþ1Þ;bðtÞ;

for all integers k.

Proof.This follows immediately from the following calculations.

Tðb^mþkðqþ1ÞÞ ¼b^mþkðqþ1Þþb^{qmþkqðqþ1Þ}

¼b^mðb^qþ1Þ^kþb^qmðb^qþ1Þ^kq

¼b^mþb^qm¼Tðb^mÞ;

sinceb^qþ1¼1.

Tððbtþb^qÞ^mþkðqþ1ÞÞ ¼ ðbtþb^qÞ^mþkðqþ1Þþ ðb^qtþbÞ^mþkðqþ1Þ

¼ ðbtþb^qÞ^mððbtþb^qÞ^qþ1Þ^kþ ðb^qtþbÞ^mððb^qtþbÞ^qþ1Þ^k

¼ ðbtþb^qÞ^mððbtþb^qÞðb^qtþbÞÞ^kþ ðb^qtþbÞ^mððbtþb^qÞðb^qtþbÞÞ^k

¼ ðt²þTðbÞ²tþ1Þ^kððbtþb^qÞ^mþ ðb^qtþbÞ^mÞ

¼ ðtþTðbÞt¹⁼²þ1Þ^2kTððbtþb^qÞ^mÞ:

Hence,

h1;mþkðqþ1Þ;bðtÞ ¼ Tððbtþb^qÞ^mþkðqþ1ÞÞ ðtþTðbÞt¹⁼²þ1Þ^{mþkðqþ1Þ1}

¼ðtþTðbÞt¹⁼²þ1Þ^2kTððbtþb^qÞ^mÞ ðtþTðbÞt¹⁼²þ1Þ^{mþkðqþ1Þ1}

¼ Tððbtþb^qÞ^mÞ ðtþTðbÞt¹⁼²þ1Þ^mþkðq1Þ1

¼ Tððbtþb^qÞ^mÞ

ðtþTðbÞt¹⁼²þ1Þ^m1¼h_1;m;bðtÞ:

The last simplification is due to the fact that ðtþTðbÞt¹⁼²þ1Þ^q1¼1 since tþTðbÞt¹⁼²þ1AFnf0g. A very similar computation shows that

h2ðtÞ ¼h2;m;bðtÞ ¼h2;mþkðqþ1Þ;bðtÞ: r It is now clear that we may restrictmto lie inf1;. . .;qþ1g. However, there is a further equivalence which shows that at most half of these values lead to distinct functions.

Lemma 3.6.For the functions f and g defined above,we have

fðtÞ ¼ f_m;bðtÞ ¼ fm;bðtÞ and gðtÞ ¼g_m;bðtÞ ¼gm;bðtÞ:

Proof.First, observe that Tðb^mÞ ¼ 1

b

m

þ 1

b^q

m

¼b^qmþb^m¼Tðb^mÞ:

Now,

Tððbtþb^qÞ^mÞ ¼ 1

ðbtþb^qÞ^mþ 1 ðb^qtþbÞ^m

¼ðb^qtþbÞ^mþ ðbtþb^qÞ^m

ðbtþb^qÞ^mðb^qtþbÞ^m ¼ Tððbtþb^qÞ^mÞ ðtþTðbÞt¹⁼²þ1Þ^2m: Thus, we have

h1;m;bðtÞ ¼ Tððbtþb^qÞ^mÞ

ðtþTðbÞt¹⁼²þ1Þ^m1 ¼ Tððbtþb^qÞ^mÞ

ðtþTðbÞt¹⁼²þ1Þ^2mm1¼h_1;m;bðtÞ: A similar calculation shows thath_2;m;bðtÞ ¼h_2;m;bðtÞ. r

We are now in a position to be able to prove our main result.

Proof of Theorem3.1. It is straightforward to verify that fð0Þ ¼gð0Þ ¼0 and fð1Þ ¼ gð1Þ ¼1. As outlined in Section 2, and in the notation of the statement of Theorem 3.1,Cis aq-clan if and only if

trace_F ðfðsÞ þfðtÞÞðagðsÞ þagðtÞÞ sþt

¼1 ð3:5Þ

for alls;tAGFðqÞwiths0t. (For, if (3.5) holds, then puttings¼0 andt¼1 shows that trace_FðaÞ ¼1.) By Lemmas 3.2 to 3.6 we need to consider trace_FððTðg^mÞ=TðgÞÞ forgAN₁ and for eachm¼1,q=2, 5 andðq1Þ=3.

If m¼1 then, forgAN₁, we have trace_FððTðg^mÞ=TðgÞÞ ¼trace_Fð1Þ. By Lemmas 3.3 and 3.4, Equation (3.5) is satisfied for all e and for all bAN₁. In this case

fðtÞ ¼t¹⁼²and theq-clan is classical by [19].

If m¼q=2 then, for gAN₁, we have trace_FððTðg^mÞ=TðgÞÞ ¼trace_Fð1=TðgÞÞ ¼1 by Lemma 3.4. By Lemmas 3.3 and 3.4, Equation (3.5) is satisfied for all oddeand for allbAN1. In this case fðtÞ ¼t³⁼⁴þt¹⁼²þt¹⁼⁴and theq-clan is FTWKB by [19].

If m¼5 then, for gAN1, we have traceFððTðg^mÞ=TðgÞÞ ¼traceFððgþg^qÞ⁴þ ðgþg^qÞ²þ1Þ ¼traceFð1Þ. By Lemmas 3.3 and 3.4, then Equation (3.5) is satisfied for alleand for allb AN1ifð5;qþ1Þ ¼1 . On the other hand ifð5;qþ1Þ01, that iseis even, then Equation (3.5) holds for only somebAN1. LetbAN1 be such that traceFð1=Tðb⁵ÞÞ ¼1 and letd¼TðbÞ²; so thatTðb⁵Þ ¼d¹⁼²ðd²þdþ1Þ. It follows thatd²þdþ100, Equation (3.5) holds, and traceFð1=dÞ ¼1 by Lemma (3.4) since ð2;qþ1Þ ¼1. Now,

Tððbtþb^qÞ⁵Þ ¼ ðbtþb^qÞðb⁴t⁴þb^4qÞ þ ðb^qtþbÞðb^4qt⁴þb⁴Þ

¼ ðb⁵þb^5qÞðt⁵þ1Þ þ ðb³þb^3qÞðt⁴þtÞ

¼d¹⁼²ðd²þdþ1Þðt⁵þ1Þ þd¹⁼²ðdþ1Þðt⁴þtÞ:

WritingA¼d²þdþ1 gives

fðtÞ ¼d¹⁼²Aðtþ1Þ

d¹⁼² þd¹⁼²Aðt⁵þ1Þ þd¹⁼²ðdþ1Þðt⁴þtÞ d¹⁼²ðt²þdtþ1Þ² þt¹⁼²

¼Aðtþ1Þðt⁴þd²t²þ1Þ þAðt⁵þ1Þ þ ðdþ1Þðt⁴þtÞ ðt²þdtþ1Þ² þt¹⁼²

¼d²ðt⁴þtÞ þd²ðd²þdþ1Þðt³þt²Þ ðt²þdtþ1Þ² þt¹⁼²: A similar calculation shows that

a¼d²þd⁵þd¹⁼² dðd²þdþ1Þ and

gðtÞ ¼d⁴t⁴þd³ðd⁴þd²þ1Þt³þd³ðd²þ1Þt

ðd²þd⁵þd¹⁼²Þðt²þdtþ1Þ² þ d¹⁼²

d²þd⁵þd¹⁼²t¹⁼²: By [6, Theorem 5],Cis the Subiacoq-clan.

Finally, letebe even, so 3jq1, and letm¼^q1₃ . Letn³ ¼gAN₁and notice that nAN₁ sinceð3;qþ1Þ ¼1. Now,

Tðg^ðq1Þ=3Þ

TðgÞ ¼g^ðq1Þ=3þg^qðq1Þ=3

gþg^q ¼n^q1þn^qðq1Þ n³þn^3q

¼

1 n²þn² n³þ_n¹3

¼ nþ¹_n2

nþ¹_n

n²þ_n¹2þ1

¼ nþ¹_n

n²þ_n¹2þ1¼ 1

nþ¹_nþ1þ 1 nþ¹_nþ1 2:

Thus,

trace_F Tðg^ðq1Þ=3Þ TðgÞ

¼0 for allgAN₁: By Lemmas 3.3 and 3.4, sinceq1

3 ;qþ1

¼1,Cis aq-clan for allbAN1. r 4 Automorphism groups of the Adelaide geometries

In this section we calculate the automorphism groups of the Adelaide geometries, starting with the Adelaide herd. First we recall some notation and definitions from [20].

LetFdenote the collection of all functions f :GFðqÞ !GFðqÞsuch thatfð0Þ ¼0.

Note that each element of F can be expressed as a polynomial in one variable of degree at mostq1 and thatFis a vector space over GFðqÞ. If fðxÞ ¼P

aixⁱAF andgAAut GFðqÞthen we write f^gðxÞ ¼P

a_i^gxⁱor, equivalently, f^gðxÞ ¼ ðfðx^1=gÞÞ^g. We will be concerned with the group PGLð2;qÞacting on the projective line PGð1;qÞ, that is,

PGLð2;qÞ ¼ fx7!Ax^g:AAGLð2;qÞ;gAAut GFðqÞg:

Lemma 4.1 ([20]). For each f AF and cAPGLð2;qÞ, where c:x7!Ax^g for A¼ a b

c d

AGLð2;qÞand gAAut GFðqÞ, let the image of f under cbe the function cf :GFðqÞ !GFðqÞsuch that

cfðxÞ ¼ jAj¹⁼² ðbxþdÞf^g axþc bxþd

þbxf^g a

b þdf^g c d

:

Then this definition yields an action of PGLð2;qÞonF,which we will call themagic action.

By the transitivity of PGLð3;qÞon ordered quadrangles in PGð2;qÞ, we can assume that a given oval has nucleus ð0;0;1Þ and contains the points ð1;0;0Þ;ð0;1;0Þ and ð1;1;1Þ. Such an oval can be written in the form

DðfÞ ¼ fð1;t;fðtÞÞ:tAGFðqÞgUfð0;1;0Þg

where f is a (permutation) polynomial of degree at most q2 satisfying fð0Þ ¼0,fð1Þ ¼1 and such that for eachsAGFðqÞthe function fswhere fsð0Þ ¼0 and fsðxÞ ¼ ðfðxþsÞ þfðsÞÞ=x,x00 is a permutation (see [11], but note that this and other references use the D notation to represent a hyperoval). Conversely, any polynomial f satisfying the above conditions gives rise to an ovalDðfÞwith nucleus ð0;0;1Þ. Such a polynomial is called ano-polynomial forPGð2;qÞ.

LetHðC1Þ ¼ fDðf_sÞ:sAGFðqÞUfyggandHðC⁰Þ ¼ fDðf_t⁰Þ:tAGFðqÞUfygg be herds. We say thatHðCÞandHðC⁰Þareisomorphicif there exists cAPGLð2;qÞ such that for all sAGFðqÞUfyg we have cf_sA hf_t⁰i under the magic action, and where the induced maps7!tis a permutation of GFðqÞUfyg. (Where, for f AF, we use hfito denote the 1-dimensional subspace ofFcontaining f.) An automor- phism of a herd HðCÞ is an isomorphism from HðCÞ to itself and the automor- phism groupAutHðCÞofHðCÞis the group of all automorphisms ofHðCÞ. In other words, the automorphism group of a herdHðCÞ ¼ fDðfsÞ:sAGFðqÞUfyggis the stabiliser offhfsi:sAGFðqÞUfyggin PGLð2;qÞ, under the magic action.

We recall the following theorem:

Theorem 4.2 ([20]). Let q¼p^e. The automorphism group of a classical herd and of an FTWKB herd isPGLð2;qÞ.The automorphism group of a Payne herd for qd32is D_2ðq1ÞcC_e.The automorphism group of a Subiaco herd for qd16is C_qþ1cC_2e.

The main result of this section is:

Theorem 4.3.The automorphism group of an Adelaide herd for q¼2^ewith ed6even andb of order qþ1is Cqþ1cC2eof order2eðqþ1Þ.

Proof. The structure of the proof is similar to that of the last part of the proof of Theorem 4.2 in [20]. Let q¼2^e where ed6 and let HðCÞ ¼ fDðf_sÞ:sAGFðqÞU fyggbe the Adelaide herd, defined as in Section 2.3 with f₀¼ f,fy¼gandaas in Theorem 3.1. Letc₁ APGLð2;qÞbec₁:x7!AxwhereA¼ TðbÞ² 1

1 0

! . Then

c₁fðtÞ ¼tf TðbÞ²tþ1 t

!

þtfðTðbÞ²Þ

¼ fðtÞ þTðb^mÞ²agðtÞ þTðb^mÞt¹⁼²A hf_TðbmÞ²ðtÞi: Similarly,

c₁agðtÞ ¼ fðtÞ þ ðTðb^mÞ²þ1ÞagðtÞ þ ðTðb^mÞ þ1Þt¹⁼²

¼ fðtÞ þ ðTðb^mÞ²þ1ÞagðtÞ þ ðTðb^mÞ þ1Þt¹⁼²A hf_ðTðbmÞ²þ1ÞðtÞi

and c₁f₁ðtÞ ¼ ðc₁fðtÞ þc₁agðtÞ þt¹⁼²Þ=a¼gðtÞ. Finally, for sAGFðqÞnf0;1g, we have

c₁f_sðtÞ ¼c₁fðtÞ þsc₁agðtÞ þs¹⁼²t¹⁼² 1þasþs¹⁼²

¼ 1þs

1þasþs¹⁼² fðtÞ þ Tðb^mÞ²þ s 1þs

agðtÞ þ Tðb^mÞ þ s¹⁼² 1þs¹⁼²

t¹⁼²

which is inhf_uðtÞiwhere u¼Tðb^mÞ²þs=ðsþ1Þ.

Consider the characteristic polynomial x²þTðbÞ²xþ1 of A over GFðqÞ. Since the roots in GFðq²Þ areb² andb^2q, andAis a root, it follows that the order of A is the order of b² in GFðq²Þ, which is the order of b in GFðq²Þ, and is qþ1 by hypothesis.

Now letc₂APGLð2;qÞbec₂:x7!Ax²where A¼ 1 0 TðbÞ² TðbÞ²

. Then

c₂fðtÞ ¼ 1

TðbÞðfðTðbÞt¹⁼²þ1ÞÞ²þ 1

TðbÞ¼Tðb^mÞagðtÞA hgðtÞi and

c₂agðtÞ ¼ 1

TðbÞðagðTðbÞt¹⁼²þ1ÞÞ²þ a² TðbÞ

¼fðtÞ þ ðTðb^mÞ²þ1ÞagðtÞ þ ðTðb^mÞ þ1Þt¹⁼² Tðb^mÞ

¼aðTðb^mÞ²þ1Þ þTðb^mÞÞ

Tðb^mÞ f_TðbmÞ²þ1ðtÞA hf_TðbmÞ²þ1ðtÞi: Finally, forsAGFðqÞnf0g, we have

c₂fsðtÞ ¼ 1

ð1þasþs¹⁼²Þ²ðc₂fðtÞ þs²c₂agðtÞ þst¹⁼²Þ

¼ s²

Tðb^mÞfðtÞ þ Tðb^mÞ þs²ðTðb^mÞ²þ1Þ Tðb^mÞ

! agðtÞ

þ s²ðTðb^mÞ þ1Þ Tðb^mÞ þs

t¹⁼²

!,

ð1þasþs¹⁼²Þ² which is inhf_uðtÞiforu¼Tðb^mÞ²þ1þTðb^mÞ²=s².

Thus the element c₂AAutHðCÞ, and has order 2e(see [20]). Nowc₂ normalises hc₁iandhc₁i V hc₂iis the identity, and we have therefore shown that the Adelaide herd admitsCqþ1cC2eas automorphism group. It is straightforward to verify that the mapx7!Ax, whereA¼ TðbÞ² 0

0 1

!

does not fixHðCÞ, and the result follows

by the maximality ofCqþ1cC_2ein PGLð2;qÞ. r

We note that, with the exception of those in the last sentence, all the calculations in the proof of Theorem 4.3 are independent of the value ofm. This proof is therefore a unified treatment of the groups of the Subiaco and Adelaide herds.

Corollary 4.4. Let q¼2^e where ed6 is even and letbAGFðq²Þbe of order qþ1.

The automorphism group of the Adelaide generalized quadrangle over GFðqÞ arising from thisbis the semidirect product ofG(in the notation of Section2.2)with the semi- direct product of a cyclic group of order q²1 and a cyclic group of order2e.Since jGj ¼q⁵,this group has order2q⁵ðq1Þðq1Þe.

Proof.See Corollary 4.1 of [20]. r

Corollary 4.5. Let q¼2^e where ed6 is even and letbAGFðq²Þbe of order qþ1.

The automorphism group of the Adelaide generalized quadrangle over GFðqÞ arising from thisb is transitive on the lines through ðyÞ. Hence there is, up to isomorphism, one associated flock and one associated translation plane.

Corollary 4.6. Let q¼2^e where ed6 is even and letbAGFðq²Þbe of order qþ1.

The automorphism group of the Adelaide herd overGFðqÞarising from thisbis tran- sitive on the ovals of the herd.Hence an Adelaide oval in PGð2;qÞis stabilised by a cyclic group of order2e.

Letq¼2^ewhereed6 is even and letbAGFðq²Þbe of orderqþ1. Suppose that e=2 is odd. Then the Adelaide generalized quadrangle is new, as follows. By com- paring the orders of the respective automorphism groups, it is immediate that if the Adelaide generalized quadrangle is not new then it is a Subiaco generalized quad- rangle. The automorphism group of the Subiaco generalized quadrangle for this qis also transitive on the lines throughðyÞ(see [30]), however, in this case the automor-

phism group of the Subiaco generalized quadrangle is not transitive on the ovals of the herd (see [22]) and this su‰ces to show that the groups are di¤erent. We remark that since in PGð2;4⁴Þthe Adelaide oval is known not to be a Subiaco oval [37], the Adelaide generalized quadrangle in PGð2;4⁴Þis new.

In a sequel to this paper, we will show that for m2G1ðmodqþ1Þ,HðCm;bÞis isomorphic toHðCm⁰;bÞif and only ifm1Gm⁰ðmodqþ1Þand hence the Adelaide generalized quadrangles are new, fored6 even. It is then immediate that the Ade- laide flocks and Adelaide planes are also new.

Let q¼2^e where ed6 is even and let bAGFðq²Þbe of order qþ1. We remark that in PGð2;qÞ an Adelaide oval over GFðqÞ arising from this b is either new or is a Subiaco oval, as follows. First, every previously known hyperoval in PGð2;qÞ where qd6 is even is either a translation hyperoval or a Subiaco hyperoval. The main theorem of [19] shows that an Adelaide oval is not a translation hyperoval, for otherwise an Adelaide herd would be either classical or an FTWKB herd; contrary to the calculation of the respective groups. Since a non-translation oval contained in a translation hyperoval has a group of order ðq1Þe, its homography group has odd order q1. Since an Adelaide oval has an induced homography group of order 2, these two ovals are di¤erent.

5 Open problems

There are several open questions and problems arising immediately from this work, as follows.

1. Is the group described in Corollary 4.6 the full stabiliser of an Adelaide oval?

The answer is known to be in the a‰rmative fore¼6 and 8 [37].

2. Are the Adelaide ovals new for all q, that is, do not belong to any previously known family? The answer is known to be in the a‰rmative fore¼6 and 8 [37].

3. For a givenq¼2^ewithed6 even, are all Adelaide generalized quadrangles iso- morphic (that is, those for di¤erentb)?

4. Do the Adelaide class of geometries and the regular cyclic class of geometries discovered by Penttila [34] form a single family?

5. Classify cyclic generalized quadrangles in characteristic 2, that is, classify herds which are stabilised by a cyclic group of orderqþ1. The known examples are the classical, FTWKB, Subiaco and Adelaide generalized quadrangles.

References

[1] L. Bader, G. Lunardon, S. E. Payne, Onq-clan geometry,q¼2^e.Bull. Belg. Math. Soc.

Simon Stevin1(1994), 301–328. MR 97b:51004 Zbl 0803.51006

[2] L. Bader, G. Lunardon, J. A. Thas, Derivation of flocks of quadratic cones.Forum Math.

2(1990), 163–174. MR 91k:51012 Zbl 0692.51006

[3] D. Betten, 4-dimensionale Translationsebenen mit 8-dimensionaler Kollineationsgruppe.

Geometriae Dedicata2(1973), 327–339. MR 49 #3652 Zbl 0272.50028

[4] W. Cherowitzo, Hyperovals in Desarguesian planes of even order. In:Combinatorics ’86 (Trento, 1986), 87–94, North-Holland 1988. MR 89c:51011 Zbl 0651.51009

[5] W. Cherowitzo,a-flocks and hyperovals.Geom. Dedicata72(1998), 221–246.

MR 99m:51011 Zbl 0930.51007

[6] W. Cherowitzo, T. Penttila, I. Pinneri, G. F. Royle, Flocks and ovals.Geom. Dedicata60 (1996), 17–37. MR 97c:51006 Zbl 0855.51008

[7] F. De Clerck, C. Herssens, Flocks of the quadratic cone in PGð3;qÞ forq small. The CaGe Reports8(1992), Computer Algebra Group, The University of Ghent, Belgium.

[8] J. C. Fisher, J. A. Thas, Flocks in PGð3;qÞ.Math. Z.169(1979), 1–11. MR 81a:05023 Zbl 0406.51004

[9] M. J. Ganley, On likeable translation planes of even order. Arch. Math. (Basel) 41 (1983), 478–480. MR 85f:51011a Zbl 0534.51005

[10] J. W. P. Hirschfeld,Finite projective spaces of three dimensions. Oxford Univ. Press 1985.

MR 87j:51013 Zbl 0574.51001

[11] J. W. P. Hirschfeld,Projective geometries over finite fields. Oxford Univ. Press 1998.

MR 99b:51006 Zbl 0899.51002

[12] J. W. P. Hirschfeld, J. A. Thas,General Galois geometries. Oxford Univ. Press 1991.

MR 96m:51007 Zbl 0789.51001

[13] N. L. Johnson, Semifield flocks of quadratic cones.Simon Stevin61(1987), 313–326.

MR 89g:51009 Zbl 0645.51005

[14] W. M. Kantor, Generalized quadrangles associated withG₂ðqÞ.J. Combin. Theory Ser. A 29(1980), 212–219. MR 82b:51007 Zbl 0465.51007

[15] W. M. Kantor, Some generalized quadrangles with parametersq²,q.Math. Z.192(1986), 45–50. MR 87f:51016 Zbl 0592.51003

[16] N. Knarr, A geometric construction of generalized quadrangles from polar spaces of rank three.Results Math.21(1992), 332–344. MR 93b:51009 Zbl 0765.51006

[17] M. Law, T. Penttila, BLT-sets over small fields II. Preprint 2000.

[18] M. Law, T. Penttila, Some flocks in characteristic 3.J. Combin. Theory Ser. A94(2001), 387–392. MR 2002i:51005 Zbl 01618730

[19] C. M. O’Keefe, T. Penttila, Characterisations of flock quadrangles. Geom. Dedicata82 (2000), 171–191. MR 2001h:51009 Zbl 0969.51007

[20] C. M. O’Keefe, T. Penttila, Automorphism groups of generalized quadrangles via an unusual action of PGLð2;2^hÞ.European J. Combin.23(2002), 213–232. MR 1 881 553 Zbl pre01740611

[21] C. M. O’Keefe, J. A. Thas, Collineations of Subiaco and Cherowitzo hyperovals. Bull.

Belg. Math. Soc. Simon Stevin3(1996), 177–192. MR 97d:51011 Zbl 0854.51006 [22] S. E. Payne, The Subiaco notebook: An introduction toq-clan geometry. Unpublished

notes. The .ps file is available on S. E. Payne’s homepage http://www-math.cudenver.edu/

~spayne.

[23] S. E. Payne, Generalized quadrangles as group coset geometries. In: Proceedings of the Eleventh Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1980), Vol. II, volume 29, 717–734. MR 83h:51024 Zbl 0453.05015

[24] S. E. Payne, A new infinite family of generalized quadrangles. In:Proceedings of the six- teenth Southeastern international conference on combinatorics, graph theory and computing (Boca Raton, Fla., 1985), volume 49, 115–128. MR 88h:51010 Zbl 0632.51011

[25] S. E. Payne, An essay on skew translation generalized quadrangles.Geom. Dedicata32 (1989), 93–118. MR 91f:51010 Zbl 0706.51006

[26] S. E. Payne, Collineations of the generalized quadrangles associated with q-clans.

In:Combinatorics ’90 (Gaeta, 1990), 449–461, North-Holland 1992. MR 94e:51010 Zbl 0767.51003

[27] S. E. Payne, Collineations of the Subiaco generalized quadrangles.Bull. Belg. Math. Soc.

Simon Stevin1(1994), 427–438. MR 95k:51013 Zbl 0803.51007

[28] S. E. Payne, A tensor product action onq-clan generalized quadrangles withq¼2^e.Lin- ear Algebra Appl.226/228(1995), 115–137. MR 96g:51014 Zbl 0841.51003

[29] S. E. Payne, The fundamental theorem ofq-clan geometry.Des. Codes Cryptogr.8(1996), 181–202. MR 97f:51013 Zbl 0874.51001

[30] S. E. Payne, T. Penttila, I. Pinneri, Isomorphisms between Subiaco q-clan geometries.

Bull. Belg. Math. Soc. Simon Stevin2(1995), 197–222. MR 96g:51013 Zbl 0842.51004 [31] S. E. Payne, T. Penttila, G. F. Royle, Building a cyclicq-clan. In:Mostly finite geometries

(Iowa City, IA, 1996), 365–378, Dekker 1997. MR 98j:51010 Zbl 0887.51003

[32] S. E. Payne, L. A. Rogers, Local group actions on generalized quadrangles.Simon Stevin 64(1990), 249–284. MR 93a:51002 Zbl 0735.51007

[33] S. E. Payne, J. A. Thas,Finite generalized quadrangles. Pitman 1984. MR 86a:51029 Zbl 0551.05027

[34] T. Penttila, Regular cyclic BLT-sets.Rend. Circ. Mat. Palermo (2) Suppl.no.53(1998), 167–172. MR 99i:51008 Zbl 0911.51005

[35] T. Penttila, I. Pinneri, Hyperovals.Australas. J. Combin.19(1999), 101–114.

MR 2001d:51010 Zbl 0939.51018

[36] T. Penttila, G. F. Royle, Classification of hyperovals in PGð2;32Þ.J. Geom.50(1994), 151–158. MR 95f:51005 Zbl 0805.51005

[37] T. Penttila, G. F. Royle, On hyperovals in small projective planes. J. Geom.54(1995), 91–104. MR 96j:51013 Zbl 0840.51003

[38] T. Penttila, G. F. Royle, BLT-sets over small fields. Australas. J. Combin. 17 (1998), 295–307. MR 99g:05043 Zbl 0911.51004

[39] T. Penttila, L. Storme, Monomial flocks and herds containing a monomial oval.J. Com- bin. Theory Ser. A83(1998), 21–41. MR 99e:51008 Zbl 0923.51004

[40] J. A. Thas, Generalized quadrangles and flocks of cones.European J. Combin.8(1987), 441–452. MR 89d:51016 Zbl 0646.51019

[41] J. A. Thas, Generalized quadrangles of order ðs;s²Þ. I.J. Combin. Theory Ser. A 67 (1994), 140–160. MR 95h:51009 Zbl 0808.51010

[42] J. A. Thas, Generalized quadrangles of order ðs;s²Þ. II. J. Combin. Theory Ser. A 79 (1997), 223–254. MR 99e:51004 Zbl 0887.51004

[43] J. A. Thas, Generalized quadrangles of order ðs;s²Þ. III.J. Combin. Theory Ser. A 87 (1999), 247–272. MR 2000g:51005 Zbl 0949.51003

[44] M. Walker, A class of translation planes.Geometriae Dedicata5(1976), 135–146.

MR 54 #8442 Zbl 0356.50022

Received 4 June, 2001; revised 16 November, 2001

W. E. Cherowitzo, Department of Mathematics, University of Colorado at Denver, Campus Box 170, P.O. Box 173364, Denver, CO 80217-3364, USA

Email: wcherowi@carbon.cudenver.edu

C. M. O’Keefe, Department of Pure Mathematics, The University of Adelaide, 5005 Australia Email: cokeefe@maths.adelaide.edu.au

T. Penttila, Department of Mathematics and Statistics, The University of Western Australia, 6907 Australia

Email: penttila@maths.uwa.edu.au