Cohen, A.M.

Wilbrink, H.A.

English

Cohen, AM Arjeh

Wilbrink, HA Henny

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The stabilizer of Dye's spread on a hyperbolic quadric in $PG(4n-1,2)$ within the orthogonal group

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Cohen, Arjeh

The stabilizer of dye's spread on a hyperbolic quadric in pg(4n-1,2) within the orthogonal group : (preprint)

Cohen, A.M. (Arjeh)

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stichting mathematisch centrum AFDELING ZUIVERE WISKUNDE (DEPARTMENT OF PURE MATHEMATICS) A.M. COHEN & H.A. WILBRINK zw 135/80 THE STABILIZER OF DYE'S SPREAD ON A HYPERBOLIC QUADRIC IN PG(4n-1,2) WITHIN THE ORTHOGONAL GROUP Preprint ~ MC JANUARI 2e boerhaavestraat 49 amsterdam P,u_n;te,d a.t .the, Ma.the.ma.Uc.al. Centlte, 49, 2e BoeJr.haave1.d1uia..t, AmJ.i.teJr.dam. The Ma.the.ma.tic.al. Centlte, 6ou.nde,d .the, 11-.th 06 FebnuaJr.y 1946, iJ.i a. non-pJto0i.t ,i.n1,.t,.i.,,tu;U.on aiming a.t .the pltomo.tlon 06 puJr.e ma.the.ma.Uc.J.i a.nd w a.ppUc.a.UoM. It iJ.i J.ipon1,01te,d by .the Ne.theJLta.nclb Gove.Jtnme,n;t .th/tough .the Ne.theJLta.nclb 0Jtga.niza.Uon 601t .the Adva.nc.e.me,n;t 06 PuJr.e Re.J.iea.Jtc.h (Z.W.O). 1980 Mathematics subject classification: 20B25, 51E20 The stabilizer of Dye's spread on a hyperbolic quadric in PG(4n-1,2) within the orthogonal group*) by A.M. Cohen & H.A. Wilbrink ABSTRACT Recently, R.H. DYE [3] constructed spreads as indicated in the title. He determin,ed their stabilizers within the relevant orthogonal group in the cases n = 2,3. The present note deals with all n ~ 3. Use is made of Holt's characterisation of certain triply transitive permutation groups 2n-1 of degree 2 + 1. KEY WORDS & PHRASES: finite classical geometry, spreads, permutation groups This report will be submitted for publication elsewhere, 1. INTRODUC'J~ION The projective space PG(4n-1,2) is viewed in the usual way as the incidence structure of 1- and 2-dimensional subspaces of the vector space 4n w2 . The hyperbolic quadric Q will be fixed as the set of projective 1 points X in PG(4n-1,2) whose homogenous coordinates (X1 ,x2 , ... ,x4n) satisfy q(X) 0. The hyperbolic quadratic form q on PG(4n-1,2) admits a symplectic polarity that we shall denote by B. A spread on the quadric Q is defined to be a 2n-1 partitioning S = {s 1 , ... ,S 2 1 } of Q into 2 +1 projective (2n-1)-2 n- +1 dimensional totally isotropic subspaces of (PG(4n-1,2) ,q). 2 . CONSTRUC'JnON OF THE SPREAD The following construction of a spread on Q is to be found in [3] Fix a nonisotropic point P and an isotropic point Q of (PG(4n-1,2) ,q) such that B(P,Q) f 0. Then the projective space H underlying P~ n Q~ is a PG(4n-3,2) with symplectic polarity B0 induced by B. By means of scalar restriction from the Galois field JF 22n-l to JF 2 , the projective line PG(1,2 2n-l) with ~ondegenerate symplectic polarity B1 can be regarded as a PG(4n-3,2) with nondegenerate symplectic polarity trace I o B1 . JF 22n-1 JF 2 Thus (H,B0 ) can be identified 2n-1 with (PG(l,2 ) , trace j o B1 ) when-JF 22n-1 JF 2 ever the latter is viewed as a projective space over JF 2 . Under this 2n-1 identification, the points of PG(l,2 ) correspond to totally isotropic (2n-2)-dimensional subspaces of (H,B0 ) partitioning H. Next, His mapped bijectively onto P~ n Q by means of projection from P. Note that totally isotropic subspaces of (H,B0 ) map into totally isotropic subspaces of (P~,qjp~) inside a, so that the partitioning of (H,B0 ) maps onto a partitioninq of P~ n Q into totally isotropic subspaces. In order to obtain a spread, note that each of these-(2n-2)-dimensional subspaces should be extended to a maximal totally isotropic subspace of (PG(4n-1,2) ,q). It follows from [2] that this can be done in precisely two different 2 ways such that no two subspaces intersect. The two resulting spreads on r2 are mapped into one another by the symmetry with center P. Moreover, the subspaces belonging to one of these two spreads are all in the same Q+ (2)-4n + orbit, where r2 4n(2) stands for the commutator subgroup of the orthogonal + group o4n(2) with respect to q. Hence, the spread is uniquely determined by the requirement that. its elements are maximal totally isotropic subspaces from a fixed Q:n(2)-orbit. The spread thus constructed will be denoted P. 3. THE STABILIZER OF THE SPREAD Let G denote the stabilizer of the spread P + within o4n(2) and let G for Ra point of PG(4n-1,2) stand for the subgroup of G fixing R. Since R 2n-1 2n-1 pff (2 ) is in a canonical way a group of automorphisms of (PG(l,2 ) , 2 trace I o B1) and thus of (H,B0 ), it can be embedded uniquely JF 22n-1 JF 2 into Gp. This implies that GP contains a subgroup K isomorphic to Prf2 (2 2n-l). The following lemma summarizes what is known about G from [3]. LEMMA. (Let q,P,K and G be as above) 2n-1 2n-1 (i) K acts on P as Prl2 (2 ) acts on PG(l ,2 ) ; (ii) GP= K - Prl2 (22n- 1 ); GP has three orbits on the set of nonisotropic points of (PG(4n-1,2) ,q) with cardinalities 1,24n-2-1, 22n-l(22n-l_l); (iii) If n 2, then G - Alt(9); (iv) if n = 3, then G The proof of (ii) can be found on page 191 in [3] in an argument that is valid in the present situation (though not explicitly stated). Statement (iv) is demonstrated by use of specific knowledge of the subgroups of Sp6 (2). The theorem which we aim to prove, shows that (iv) is representative for what happens for n ~ 3. THEOREM. Let n ~ 3. Suppose Pis a nonisotropic point and Q an isotropic point of a nondegenerate hyperbolic space_ (PG(4n-1,2) ,q) such that P + Q is a hyperbolic line. Let P be the spread constructed in 2 departing from P and Q, and let G be as defined in 3. Then G =GP= pff2 (2 2n-l). 4. PROOF OF THE THEOREM We proceed in four steps. (4.1) G does not possess a normal subgroup which is regular on the set of nonisotropic points of (PG(4n-1,2) ,q). PROOF. Suppose N is a counterexample. Then GP acts on N by conjugation as it does on the nonisotropic points. In particular N has two GP-orbits distinct from {1}. Let p and q denote the orders of representatives from a b these two orbits. Then by Cauchy's lemma N has order p q for a,b E JN; moreover p and q are prime numbers. On the other hand, the regularity of 3 N implies that its order is 22n-l(2 2n-1). The comparison of these two expressions for jNj yields that 22n-1 is a prime power, which is absurd.~ (4.2) If N is a nontrivial normal subgroup of G, then [G:N] = [GP:NP] is a divisor of 2n-1. PROOF. If G = GP, the statement concerns G = Prl2 c2 2n-l) and is known to hold. So we may assume G > G for the rest of the proof. In view of the p orbit structure of GP described in (ii) of the lemma, this means that G is primitive on the set of nonisotropic points. So any nontrivial normal 2n-1 2n subgroup N of G is transitive on these 2 (2 -1) points, so 2n-1 [G:N] = [GP:NP]. Moreover NP is normal in GP= Prl2 (2 ) , whence NP= 1 or we are through. The former possibility, however, is excluded by (4.1) ~ (4.3) The permutation representation of G on Pis faithful. PROOF. Let N be the kernel of this representation. If N is nontrivial, then = [G :N] by (4.2); but (i) of the lemma states p p that NP= 1, whence [G:N] [G:N] = JG I, contradicting (4.2). The conclusion is that N is trivial. p (4.4) If n ~ 3, then G = Gp. PROOF. By (4.3) the group G can be regarded as a triply transitive ' f d 22n-l 1 1· · f permutation group o egree +. App ication o a theorem by Holt [4] yields that G contains a normal subgroup N isomorphic to either ~ 4 2n-1 2n-1 2n-1 Sym (2 + 1), Alt (2 + 1) or Psl2 (2 ) • Comparing orders with I 2n-1 jG, we obtain-that N is an isomorph of Psl2 (2 ) . From (4.2) it follows that G = Gp 181. 5. REMARKS For n = 2, the arguments of the proof are equally valid. They result in: G = Prl2 (2 2n-l) or G = Alt (9). Together with the observation that all spreads are in a single o:n(2)-orbit, this reestablishes (iii) of the lemma. De Clerck, Dye and Thas [1] have shown that any spread leads to a partial geometry with parameters (s,t,a) = (2 2n-l_l, 22n-l, 22n-2 ) on the nonisotropic points of PG(4n-1,q). Using the above theorem, it is not hard to see that G is the part of the automorphism group of the partial geometry derived from P that is contained in o:n(2). 6. REFERENCES [1] DE CLERCK; DYE,R.H.; THAS, J.; preprint ~ [2] DIEUDONNE· J; La geometrie des groupes classiques, [3] DYE, R.H.; Partitions and their stabilizers for line complexes and and quadrics, Annali di Matematica pura ed applicata (4) 114 (1977), 173-194. [4] HOLT, D.F.; Triply-transitive permutation groups in which an involution central in a Sylow 2-subgroup fixes a unique point, J. London Math. Soc (2) 15 (1977), 55-65. 

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 The stabilizer of Dye's spread on a hyperbolic quadric in$PG(4n-1,2)$ within the orthogonal groupCitation for published version (APA):Cohen, A. M., & Wilbrink, H. A. (1980). The stabilizer of Dye's spread on a hyperbolic quadric in $PG(4n-1,2)$within the orthogonal group. Atti della Accademia Nazionale dei Lincei. Serie Ottava. Rendiconti. Classe diScienze Fisiche, Matematiche e Naturali, 69(1-2), 22-25.Document status and date:Published: 01/01/1980Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 16. Nov. 2023

 The stabilizer of Dye's spread on a hyperbolic quadric in$PG(4n-1,2)$ within the orthogonal groupCitation for published version (APA):Cohen, A. M., & Wilbrink, H. A. (1980). The stabilizer of Dye's spread on a hyperbolic quadric in $PG(4n-1,2)$within the orthogonal group. Atti della Accademia Nazionale dei Lincei. Serie Ottava. Rendiconti. Classe diScienze Fisiche, Matematiche e Naturali, 69(1-2), 22-25.Document status and date:Published: 01/01/1980Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 04. Oct. 2023

The stabilizer of Dye's spread on a hyperbolic quadric in $PG(4n-1,2)$ within the orthogonal group

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The stabilizer of Dye's spread on a hyperbolic quadric in PG(4n−1,2)PG(4n-1,2)PG(4n−1,2) within the orthogonal group

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The stabilizer of Dye's spread on a hyperbolic quadric in $PG(4n-1,2)$ within the orthogonal group