Cooperstein, B.N.

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A finite flag-transitive geometry of extended G2-type : (preprint)

CWI's Institutional Repository

stichting mathematisch centrum AFDELING ZUIVERE WISKUNDE (DEPARTMENT OF PURE MATHEMATICS) zw 176/82 B.N. COOPERSTEIN A FINITE FLAG-TRANSIT I VE GEOMETRY OF EXTENDED G2-TYPE Preprint ~ MC AUGUSTUS kruislaan 413 1098 SJ amsterdam blt>LlU ir1LU~ fv1/, 1HEfv1A TISCH CENT HUM -M/SfEFiDAM--PJunted a.t .the Ma.thema.tic..a.l C enttLe., 41 3 K.!U.U6la.a.n, Aw.,.teJr.dam. The Ma.thema:tlc..ai. CenttLe , E,ou.nde.d .the 11-.th o!, Febll..WVl.y 1946, ,l6 a. n.on-plLo E,U ,i,n&t:.li.u.il..o n a.i.m[n.g a.t .the plLomotio n o 6 pu.JLe ma.thema.tiC-6 and U6 a.ppUc..a.tianJ.i. I.t ,l6 1.,pon&oJLed by .the Nethe.JLla.n.dl., GoveJLnment .thMu.gh .the NetheJcla.nd1., OJr.ga.n,i,za.tion !,oJL .the Adva.nc..ement o O Pu.JLe Ru e.Mc..h ( Z .W. 0.) • 1980 Mathematics subject classification: 5 IE30 A finite Flag-Transitive Geometry of Extended G2-Type*) by Bruce N. Cooperstein**) ABSTRACT The pulrpose of this note is to give an explicit construction of a finite flag-transitive GAB with an extended G2-diagram having the group G2(3) as automorphism group. In our example it will be apparent that the intersection property is satisfied. KEY WORDS & PHRASES: Buekenhout-Tits geometries, geometries of extended G2 type, ChevaZZey group G2 (3) *) This report will be submitted for publication elsewhere. **) University of California, Santa Cruz, California, 95064 0. INTRODUCTION Geometries that are almost buildings (GABs) were introduced by TITS in [II]. They are BUEKENHOUT-TITS geometries [1] in which all rank two residual geometries are generalized polygons, except they need not satisfy the in-tersection property. Tits has shown that they exist in great number, in-cluding finite ones. In [7] KANTOR remarks that the situation for finite GABs with large automorphism groups, other than those arising from buildings appear to be rare. In [7] KANTOR briefly describes four finite GABs having flag-transitive automorphism groups. The only other known finite flag-transitive GAB was constructed by RONAN and SMITH [9] from the Suzuki spo-radic group. The purpose of this note is to give an explicit construction of a finite flag-transitive GAB with an extended G2-diagram having the group G2 (3) as automorphism group. In our example it will be apparent that the in-tersection property is satisfied. I. GEOMETRIES OF EXTENDED G2-TYPE We will be concerned here with incidence structures I= (P,L,TI;I) with three types of objects: P, whose elements are called points; L, whose elements are called lines; and TI' whose elements are called planes., together with a symmetric relation I on E = P u L u TI. We set 0 = {P,L,TI}. Suppose {X,Y,Z} = 0, x e X. Set Y = {ye Ylx I y}, Z = {z e Z Ix Iz}, X X and Ix = Ily u Z . X X (Y ,Z ;I) is called the residue of I at x, I . X X X X We say 1 is of extended G2-type or belongs to the diagram o-o=:o if the following are satisfied 2 (i) If X e: 0 ' X ::/:ye: X, then x and y are not incident; (ii) If X € p~ I is a generalized hexagon; X (iii) If ,e_ e: L, I,e_ is a complete bipartite graph; and (iv) If lf e: IT' I is a projective plane.· lf In our construction all residues will have order two. 2. THE GEOMETRY OF NON-ISOTROPIC POINTS IN THREE DIMENSIONAL UNITARY SPACE Let K = lF 2, <-r> = Gal (K/k) , k = lF • Denote images under -r by - • Let q q V be a three-dimensional vector space over Kand h: vxv--+ Ka non-degener-ate hermitian form, that is, h should satisfy (i) For v e: V, w--+ h (w) = h(w,v) is linear V (ii) For v,w € v, h(w,v) = h(v,w) (iii) h = 0 if and only if V = o. V (V,h) is a unitary space over K. Let N = {<v>lv e: V, h(v,v) ::/: O}. N is the set of non-isotropic points of (V,h). Define a graph on N as follows: for p ::/: q e: N, p ~ q if and only if h(p,q) = O. Let A be the collection of maximal cliques in (N,~). Clearly these are triples. We collect some facts about the partial linear space (N,A). (2. 1) 2 2 !NI= q (q -q+l) This is easy: IPG(V)l = q4 + q2 +I.The number of isotropic or absolute 3 points (<x> is absolute if h(x,x) = O), is q + I. F.or x e: N, set r(x) = {y e: N: x ~ y}. We let d(,) = dr(,) be the usual metric associated with (N,r) and r (x) be the points at distance t from x. . t (2.2) lr(x)I 2 = q -q • .L This is again easy: x = < v e:V: h(x, v) = 0 > is a non-degenerate two space, so IPG(x.L)I = q2 + I. The number of absolute points in x.L is q + I. Set A = {A e: A: x e: A}. Then clearly from (2.2) we have X (2.3) Next suppose x f y. There are clearly two possibilities: (i) <x,y> is non-degenerate; and (ii) Rad <x,y> = <x,y>.l n <x,y> is a (isotropic) point. In case (i) we see that x.l n y.l is a single point in N, while in (ii) .l Rad Th · ( ·) d ( ) 2 and 1·n case (1· 1·) <x,y> = <x,y>. us in case 1, r x,y = dr(x,y) ~ 3. A simple count yields (2 .4) 3 Now it is not difficult to see that O(V,h) = {T € GL(V): h(Tv,Tw) = h(v,w)} acts transitively on pairs {x,y} ~ N with d(x,y) = 2, and on pairs with Rad <x,y> IO. It therefore follows that (2.5) (N,r) is distance transitive with diameter 3. We now turn our attention to pairs {x,y} with Rad <x,y> IO. Let x = <v>, y = <w> where h(v,v) = h(w,w) = I. Suppose :\ = {x =x 1 ,x2,x3} is a ·1ine on x. Let v=v1, x. = <v.> where h(v.,v.) = I, i = 2,3. Now set 1 1 1 1 R = Rad <x,y>. Then w =av+ r where r € R. Sinceh(w,w)= h(v,v) = I, with-out loss of generality we may assume a= I. Since h(v,r) = 0 and .l .l v = x = <x2,x3>, there are b,c € K so r = bv2 + cv3 • Since h(r,r) = 0 we must have (2.6) bb +cc= O. We will determine conditions for d(y,x.) = 2 for i = 2,3. Now d(y,x.) = 2 1 1 if and only if y.l nx7 € N. x7 = <x 1,x.> where {i,j} = {2,3}. Note if 1 1 J a. v 1 + S "j € y.l , then a. S I O, so we may take S = I • If h(nv1+vj ,v1+bv2+cv3 ) = 0 then a.={-~ - C j = 2. j = 3 Set j = 2 J = 3 Now h(a.v1+v.,a.v1+v-.) = a.a. + I. Now if we assume char (k) :f: 2, then by J J ,J J J J (2.6) we cannot have a2 a2 +I= 0 = a3 a3 +I.We have therefore shown (2. 7) If d(x,y) = 2, A E J\.x' then An r 2 (y) :f: 0. Now if q > 3 we can easily see that there are pairs x,y with d(x,y) = 3 and lines A on x such that A -{x} =.. r2 (y). However, from (2.6) we see that (2.8) We have thus demonstrated all we need for 4 (2.9) THEOREM. The geometry (N,J\.) is a generalized hexagon if, and only if, q = 3. (2.10) REMARK. When q = 3 the generalized hexagon (N,A) is the dual of the usual (2,2)-generalized hexagon associated with G2 (2). [The usual G2(2n) hexagon is the one embeddable in PG(5,2n)J. This follows from the fact that if Q = o2 (Gx),x EN, then Q', the com-mutator subgroup of Q, has order two. 3. THE OCTAVES AND THE G2-GENERALIZED HEXAGON Let k be a commutative field, ©(k) =©the split octaves over k. © is a composition algebra, that is© is an algebra with identity and admits a non-degenerate quadratic form Q such that Q(x.y) = Q(x) Q(y). We can find an othonormal base I= e0 ,e 1, ••• ,e7 for®, with e0 the identity element such that multiplication in© is determined by (3. I) and 2 e. = - I , 1 ~ i ~ 7 (3.2) eiej = - ejei = ek whenever (ijk) is one of the three cycles (l+r,2+r,4+r) where i,j,k,r run through the integers modulo seven and take their values in {1,2, ••• ,7}. Let W = e~ = <e 1, ••• ,e7>, so that QIW is a non-degenerate quadratic form (with maximal Witt index). Let G = Aut(~). Clearly Gleaves e0 and W invariant. 5 It is well-known (cf.[S])that G is the Chevalley group G2(k). Moreover, it is well known that there is, up to isomorphism, only one such algebra ~( cf.(3.1) in [5]). Consequently we have (3.3) If f 1, ..• ,f7 satisfy (3.1),(3.2) then there exist a E Aut(IO) with of . = e. , 1 ::;; i ::;; 7. 1 1 Of interest to us will be the set~= {{e.,-e.},1::;; i::;; 7 }, and its full 1 1 stabilizer, G~, in G. This group is also well-known (see [3] and [4]): (3.4) G~ is a non-split extension of an elementary abelian group E of or-der 8 by PSL3(2). For the sequel we assume that -1 is not a square ink. Then K -· <eO,e 1> = k(e 1) ~ k [t]/(t2+l) is a quadratic extension of k. J. Set V = <e2 , ••• ,e7> = W n e 1• Note that V becomes a three dimensional vector space over K by restriction of the multiplication of~ to KxV. For u = aeO + be 1 EK, set u = aeO - be 1. Then - generates the Galois group of Kover k. Next define h: vxv--+ K to be poµ. lvxv, whereµ is the multiplication of© and pis the projection of© onto K. Then (3.5) his a non-degenerate hermitian form on V with associated automorphism -. Moreover, a E G if and only if crlv is a unitary transformation, i.e. <el> preserves h. 4. THE CONSTRUCTION OF THE EXTENDED G2 - GEOMETRY OF ORDER 2 We retain the notation of the previous sections. Further we set p = <el> G {<w> = : W E W, Q(w) = 1} L = ,e_G where£.= {<el>,<ez>,<e4>} ; and , II G where { <e. >: 1 ~i ::;;7}. = 'If 'If = 1 Let I be synnneterized inclusion of subsets of P restricted to E =Pu Lu II. We will prove (4.1) THEOREM: I= (P,L,IT;I) is a geometry of extended G2-type if and only if k = 1F3 • We proceed to prove this in a series of steps. Note that G = Aut(G>) is flag-transitive by (3.3).Because of this it suffices to check the re-sidues I , I O , I7f. <el> ,{... (4.2). I is a projective plane of order 2. 7f 6 pf: {<e.>, <e.>, <ek>} E L7f for all triples (ijk) as in (3.2). If these are - l. J all the lines in L7f, then the result is clear. Now C = G~ ~ G7f and is two transitive on 7f. Moreover, for any i ~ j C fixes <ek> where (ijk) . <ei>,<e,> is as in (3.2) and is transitive on 7r-{<e.>,<e.>,~ek>}. It therefore suf-1. J fices to prove {<e 1>,<e2>,<e3>} / L. But this is obvious since G tra Land elements of G preserve multiplication. (4.3) I is a generalized hexagon if and only if k = JF3 • <el> i pf: Let l EL . Define 0(l) = l n e 1. Now l EL if and only if the sub-<e > <e 1> algebra generat!d by eO and l is k eO + k l, and this occurs if and only if 0(l) is a one dimensional non-isotropic subspace of the unitary K-space V = W n ei1• Thus 0 (L ) = N (V, h), the non-isotropic points .. Because of the <e > flag transitivity we s!e that l 1,l2 EL lie in a cormnon plane if and <e > only if the points 0(l1), 0(l2) of V are &rthogonal. Thus I is isomorphic <e > to the geometry (N,A) of section two. By (2.10) I is a geJeralized hexa-<e1> gon if and only if k = ]F 3 , and in this case I is the dual of the usual <el> · (2,2)-hexagon. (4.4) If k = 1F3 then I..e_ is the complete bipartite graph K3 , 3 • pf: Clearly Il is a complete bipartite graph and as Ill = 3 it suffices to prove that there are three planes containing l. However, if 7f' E Ill , then 7f 1 E II< and 7f 1 is a line of I containing l. Since I has order e 1> <e > <e 1> (2,2) it follows that there are prJcisely three planes containing l. This completes the proof of the theorem. 5. CONCLUDING REMARKS Following KANTOR [4] we define an apartment to be a subset 6 of P such that there are automorphisms r,s,t of I leaving 6 invariant such that r 6 , s6 , t 6 satisfy the relations •-•=•while <r,s,t>6 acts flag-transitively. We have 7 (5. 1) PROPOSITION. Apartments do not exist. PROOF. Suppose that t.,r,s,t exist •. Then there is a point PO E t. fixed bys and t and points P1, ..• ,P6 int. all collinear with PO such that Pi is collinear with P. 1(modulo 6). Moreover triangles int. are not lines of I. t. t. i+ . However, <s ,t > contains an elementary group of order 4 which cannot act regularly on {P1, .•. ,P6 }, therefore there is an involution Tin <s,t> fixing at least two of {P1, ... ,P6}. But then since T fixes PO,IFixt.(T)l2: 3. However, for any involution in G, Fixt. (T) is a line. This contradicts the fact that t. contains no full lines. (5.2) REMARK. In [6] Goldschmidt studied groups G generated by a pair of sub-groups P1,P2 where P 1,P2 contain a common 2-Sylow S of G and IP.:SI = 3, i = 1,2, and determined all such groups. In analogy with the i groups of Lie type we might call such subgroups P. parabolics. This theory i has been extended to the case IP 1 :SI = q + I, P/O2(Pi) === L2 (q). CHERMAK [2], TIMMESFELD [IO] and others have considered the problem of determining groups G = <P 1, ... ,Pn> with n 2: 3 such that all Pi contain a common 2-Sylow S, IP.:SI == q + I and P./O2(P.) - PSL2 (q), q a power of two, especially the i i i case where for any i # j,<Pi,Pj>/O2(<Pi,Pj>) === L2 (q) xL2 (q) or L3 (q) or SL3 (q). In general the resulting group is a Chevalley group of type A ,D ,E . n n n over a field of characteristic two. Our construction provides an example of a group G generated by three "parabolics" P1,P2 ,P3 two Sylow S with IP.:SI = 3 and such that for any i # i containing a common j,<P.,P.>/O2(<P.,P.>) i J i J is one of PSL3 (2), PSL2 (2)xPSL2 (2), G2(2). This suggests that a general clas-sification will prove extremely difficult. (5.3) FINAL REMARK. In some sense the existence of such a geometry for G2 (3) should not be surprising: A result of MASON'S [8] classifies the groups G of characteristic 2,3-type under very mild hypotheses. The groups are PSP4 (3), ULf(3) and G2 (3). Of course PSP4 (3) is not particularly exceptional because of the isomorphism PSP4 (3) === Q~(2). In [7] KANTOR constructs a GAB for u4 (3), and our construction is for G2(3). Thus these groups act flag-transitively on 1F 3 -buildings and as flag-transitive groups on IF 2 - 'near' buildings. 8 ACKNOWLEDGEMENT I would like to thank Dr. Arjeh M. Cohen for suggesting the use of the octave algebra for the construction and aiso for many helpful conversations. I would also like to thank Peter Cameron who pointed out that apartments do not exist. REFERENCES [ I J BUEKENHOUT, F. , Diagrams for Geometries and Groups., J of Comb. Theory A 27(1979), 121-151. [2] CHERMAK, A., On Certain· Groups with Parabolic Type Subgroups over zz 2 , .J. of London Math Soc.(2), (1981), 265-279. [3] COHEN, A.M., Finite Groups of Real Octave Automorphisms., M3.th. Centre Report ZN 95/80, 1980 Amsterdam. [4] COXETER, H.S.M., Integral Cayley Numbers., Duke Math J. 13(1946), .561-578. [5] FAULKNER, J. R. & J.C. FERRAR, Exceptional Lie Algebra.s and Related ,tlgebraic and Geometric Structures., Bull. London Math Soc., 9 (1977), 1-35. [6] GOLDSCHMIDT, D., Automorphisms of Trivalent Graphs., Ann. of Math (2), 111, (1980), 377-406. [7] KANTOR, W.M., Some Geometries that are almost buildings. [8] MASON, G., Two Theorems on Groups of Characteristic 2-type., Pac . .J. of Math. Vol. 57, No.I, (1975), 233-253. [9] RONAN, M. & S. SMITH, 2-Local Geometries for some sporadic Groups., Proceedings of Symposia in Pure Mathematics., Vol. 37, ( 1980), 283-289. [10] TIMMESFELD, F.G., Tits Geometries and Parabolic Systems in Finite Groups., to appear. [II] TITS, J., A Local Characterization of Buildings, to appear. MC NR 35222 

A finite flag-transitive geometry of extended G2-type : (preprint)

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