Adv. Geom.2(2002), 197–200 Advances in Geometry (de Gruyter 2002
Note on span-symmetric generalized quadrangles
William M. Kantor*
(Communicated by T. Grundho¨fer)
Abstract. We determine all span-symmetric generalized quadrangles of orderðs;tÞ for which t<s2.
In a generalized quadrangleQof orderðs;tÞwiths;t>1, a lineLis called anaxis of symmetryif the groupTðLÞof all automorphisms (‘‘symmetries’’) that fix every line meetingLhas the maximal possible orders. Moreover,Qis calledspan-symmetricif there are two disjoint axes of symmetry. These notions were introduced in [6] and [10]
in view of the known examples (Qð4;qÞand Qð5;qÞ, arising respectively from quadrics in 4- and 5-dimensional projective spaces). In this note we will prove the following Theorem.Any span-symmetric generalized quadrangle for which t0s2is isomorphic to Qð4;sÞ.
The hypotheses provide two disjoint axes of symmetry,LandM, and hence also the groupG¼hTðLÞ;TðMÞiof automorphisms they generate. The proof of the theorem is an elementary combination of the classification of 2-transitive permutation groups in which the stabilizer of a point has a normal subgroup regular on the remaining points ([4], [9]), standard results about central extensions of such groups ([8], [1], [3]), and the fact thatjGj ¼ ðsþ1Þsðt1Þ([10, IV.2], [7, 10.7.3]) (proved combinatorially using eigenvalue techniques!).
The cases<tof the theorem was announced long ago [5] and mentioned in ([10, p. 88], [7, p. 225]); the simplification and variation of that proof in the special case s¼twere noticed one week after [5]. The proof given below is straightforward, and hence was never published. Publication at this point stems from the need for the im- possibility ofs<t<s2 in lovely new results of K. Thas on span-symmetric general- ized quadrangles [12]. The case s¼tof the theorem was also obtained by him [11], independently, using the exact same results ([4], [9], [8], [1], [3]), but handling the sharply 2-transitive possibility di¤erently (employing geometric results in place of elementary group theory).
* This research was supported in part by the NSF.
Proof of the Theorem. By [7, pp. 224–225],sct, the orbitLG¼ fL0;. . .;Lsghas size sþ1, andG¼hTðLÞ;TðMÞiacts on this set as a 2-transitive permutation group of just the sort we noted above was classified in [4] and [9] (here TðLÞis the required normal subgroup of the stabilizer GL of the line L). Thus, if Kis the kernel of this action, thenG=K is one of the following: (i) PSLð2;qÞ,s¼q; (ii) PSUð3;qÞ,s¼q3; (iii) a Suzuki group SzðqÞ,s¼q2; (iv) a Ree group RðqÞ,s¼q3; or (v) a sharply 2- transitive group. We will view the casesqc3 of (i) andq¼2 of (ii) and (iii) as lying in case (v), so thatG=Kis a simple group in (i)–(iv) unlessG=KGRð3ÞGPGLð2;8Þ. The groupsKandTðLÞnormalize one another, and hence½K;TðLÞcKVTðLÞ, whereKVTðLÞ ¼1 by ([10, p. 85], [7, p. 225]), here½K;TðLÞ:¼hk1u1kujkAK;
uATðLÞi. Thus,Kcommutes with eachTðLiÞand hence is contained in the center ZðGÞ of G. Consequently, K ¼ZðGÞ since ZðG=KÞ ¼1. Note that jGLM=Kj jKj ¼ jGLMj ¼ jGj=ðsþ1Þs¼t1.
In (i)–(iv) we claim thatGequals its derived groupG0 if we (temporarily) exclude the case G=KGRð3Þ. For, ðG=KÞ0¼G=K, so that G¼G0K. If t¼s then ðjTðLÞj;jKjÞdividesðs;t1Þ ¼1, so thatTðLÞcG0and henceG¼G0. For general t we use the structure of GL=K in order to show that TðLÞcG0: in each of the groups we are considering in (i–iv), TðLÞK=KGTðLÞ lies in ðGL=KÞ0. Since the actions ofGL onTðLÞandTðLÞK=K are equivalent, it follows thatTðLÞcG0 and hence thatGcG0for anyt, as claimed.
Consequently,Gis a group such thatG¼G0andG=ZðGÞis one of the groups in (i)–(iv). The references ([8], [1], [3]) obtain the unique (up to isomorphism) largest group H¼H0such that H=ZðHÞis isomorphic to one of the groups in (i)–(iv), so thatGGH=H0 for someH0cZðHÞ.
With this preparation, we can now consider the individual cases (i)–(v).
(i) Here GGPSLð2;qÞ or SLð2;qÞ, unless q is 4 or 9 and GG2:PSLð2;4Þ, 3:PSLð2;9Þ or 6:PSLð2;9Þ [8, p. 119]. Since q¼sct and jGj ¼ ðqþ1Þqðt1Þ it follows thatGGSLð2;qÞands¼t: the possibilitiess¼4 andt1¼23, as well as s¼9 andt1¼34 or 64, are all eliminated by the standard divisibility condition ðsþtÞ jstðsþ1Þðtþ1Þ[PT2, 1.2.2]. The subgroupsTðLiÞare uniquely determined as the Sylow subgroups ofGfor the prime dividingq. SinceQis uniquely reconstructible fromGand theTðLiÞ([6, p. 235], [7, p. 227]),Qis as stated in the theorem.
(ii) HereGGPSUð3;qÞor SUð3;qÞ[3] and s¼q3, so thatt1¼ jGj=ðsþ1Þsis ðq21Þ=3 orq21<s1, a contradiction.
(iii) Here GGSzðqÞ, 2:Szð8Þ or 22:Szð8Þ [1] and s¼q2, which produce the con- tradictiont1¼ jGj=ðsþ1Þsc4ðq1Þ<s1.
(iv) If q03 then GGRðqÞ [1] and s¼q3 produce the contradiction t1¼ jGj=ðsþ1Þs¼q1<s1.
Suppose thatq¼3. ThenG=K has a normal subgroupS=KGPSLð2;8Þof index 3, and jTðLÞVSj ¼ jðTðLÞVSÞK=Kj ¼9. We can apply an earlier argument to the subgroup H generated by the G—conjugates of TðLÞVS: we have HK=K ¼S=K, TðLÞVH ¼TðLÞVS,TðLÞVHGðTðLÞVHÞK=KcðHLK=KÞ0and henceTðLÞV HcH0. Then H ¼H0 and H=ZðHÞGPSLð2;8Þ, so that ZðHÞ ¼1 and HG PSLð2;8Þ by [Sch]. Since H is transitive on the G—conjugates of TðLÞVH it is transitive on the conjugates ofTðLÞ, so thatHTðLÞcontains all such conjugates and
William M. Kantor 198
hence is G. Now jGj ¼ jHj jTðLÞ:TðLÞVHj ¼ jRð3Þj produces the same contra- diction as before.
(v) This and (iv) with q¼3 are the only cases requiring some e¤ort. Here sþ1¼pe for some prime p, and there is an elementary abelian normal subgroup N=K of orderpe. SinceKcZðNÞ,Nis nilpotent and hence has a unique Sylow p- subgroup P. Since P is transitive on LG¼ fL0;. . .;Lsg, the group hP;TðLÞi¼ PTðLÞcontains all of the groupsTðLiÞand hence is justG. Thus,KcP.
SincejG=Kj ¼ ðsþ1Þswe havejKj ¼t1. We may assume thatt>3 [7, Ch. 6].
Ifs¼tthens1 andsþ1 are both powers of p, so thats1c2, which is not the case.
This concludes the proof when s¼t. It remains to derive a contradiction when s<t. Clearly, P0cK. Maschke’s Theorem [2, pp. 66, 177] implies that P=P0¼ ðK=P0Þ ðB=P0Þ for some subgroup B normalized by TðLÞ and hence also by PTðLÞ ¼G. As above, it follows that G¼hB;TðLÞi¼BTðLÞ and hence that K=P0¼1, so thatP0¼K¼ZðGÞ.
Let xAPK. For any y;zAPwe have½x;yz ¼ ½x;y½x;z(cf. [2, p. 18]). Thus, A:¼ f½x;y jyAPg is a subgroup of ½P;P ¼P0¼K. Here, A depends only on the coset xK of x, while ½x;yK ¼ ½x;y for any y and ½x;K ¼1, so that jAjc jP=Kj 1¼s. The 2-transitivity ofG=K implies thatGL acts transitively (by conju- gation) on the set of nontrivial cosetsxKofKinP, while centralizingKand henceA.
Thus,Ais the same for each such cosetxK, and henceA¼ ½P;P ¼K.
Now jKj ¼ jAjcs<t¼ jKj þ1, and hence t¼sþ1, whereassþtmust divide
stðsþ1Þðtþ1Þ. r
Acknowledgment. I am grateful to Hendrik Van Maldeghem for stimulating the writing of this note.
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Received 2 April, 2001
W. M. Kantor, Department of Mathematics, University of Oregon, Eugene, OR 97403, U.S.A.
Email: [email protected]
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