J. B. F I N K

FLAG-TRANSITIVE

PROJECTIVE

PLANES

1. I N T R O D U C T I O N

The purpose of this paper is to provide a proof of the following: T H E O R E M . Let G be a collineation 9roup, transitive but not regular on the flags of the projective plane rc of order n. I f n is not a fourth power, then ~z is Desarguesian and G contains the little projective 9roup of re. The odd-order version of this theorem appears in Dembowski [2, 4.4.14] and is attributed to J. McLaughlin. It should be noted that if G is flagregular, then n is even, and n 2 + n + 1 is a prime (see [5]). The only examples of such planes known to the author are those of order 2 and 8, both Desarguesian. The author is indebted to J. McLaughlin and U. Ott, who inspired m a n y of the ideas in this paper.

2. N O T A T I O N

AND BASIC RESULTS

For the main definitions and notation we refer the reader to Dembowski [2]. If X and Y are sets, then I XI denotes the cardinality of X, and X \ Y denotes the set of elements of X not in Y. If G is a collineation group of the plane zc, and xl, x2, ... are elements of re, then we denote by Gxl ' x~.... the subgroup consisting of those collineations in G leaving each xl, x2 . . . . fixed. If G preserves the subplane go of re, and K is the kernel of the action of G on rco, then the factor group G/K is denoted G ~°, called the group induced by G on n0. If H ~ G, we denote by ( H ) the subgroup generated by H, and by iF(H) the configuration in z~ whose elements are left fixed by each collineation in H. Note that iF(H) = iF((H)), and that iF(A) n iF(B) = iF(A u B). If K is a subgroup of G, we write K < G and denote the index of K in G by [ G : K [. If IF(K) is a subplane of n, we call K planar. If K is maximal in G with respect to being planar, we say that K is plane-maximal in G. We shall have occasion to use the following results. Although m a n y of these hold in much more general settings, for our purposes it will suffice to assume that ~ is a finite projective plane of order n and that G is a group of collineations on re.

Geometriae Dedicata 17 (1985) 219-226. 0046-5755/85/0173-0219501.20. © 1985 by D. Reidel Publishing Company.

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I. Let H < G. I f x is an element of n, then the H-transivity class determined by x has cardinality I H : H ~ I. The normalizer of H in G permutes the elements of ~(H). As a corollary to this, we have: II. If G is transitive on the flags of re, then for each flag (P, l), we have (i) IG:Gel = [G:Gll = n 2 + n + 1. (ii) ]Ge:Ge.~] ---IG~:Ge.~] = n + 1. III. (See I-6].) I f G is transitive on the points (or lines) of rc and contains a nontrivial perspectivity, then zr is Desarguesian and G contains the little projective group of re. IV. (See [1].) I f c~ ~ G is an involution, then one of the following holds: (i) ~:(~) is a subplane of order x/-£. (ii) o" is a perspectivity. In case (ii), a will be a homology if n is odd and an elation if n is even. V. (See [4].) Let S < G be a 2-group. I f S is planar, then the order of ff:(S) is a 2r-root of n, called a 2r-root subplane of m VI. (See [3].) For i = 1, 2, let ai be an involutory (ci, l~) homology. I f cl is on 12 , then trla 2 is an involutory (l~12, clc2) homology. VII. Let a ~ G fix all the points on l in re. If a fixes more than one point off l, then ~ = 1, the identity collineation. For the odd-order version of the theorem, we shall be particularly interested in applying these last four results to elementary abelian 2-groups of collineations. For the remainder of this section, assume that ~ has odd order and that P, Q, and R are three distinct noncollinear points. Crucial to our discussion will be the following application of Results VI and VII. L E M M A 1. Let a be an involutory (P, QR)-homology and z be an involutory (Q, PR)-homology. Then at is the only involutory (R, PQ)-homology that zt admits. Proof. Let p be an involutory (R, PQ)-homology. Then p(trz) = (pcr)z fixes all the points on PQ and PR, so must be the identity. Hence, p = at. [] We shall use A(P, Q, R) to denote the group (a, z) of Lemma 1. The group A(p, Q, R) is an elementary abelian group of order 4 whose fixelement configuration is precisely (P, Q, R ) . That these two properties characterize this group is a consequence of the following:

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L E M M A 2. Let A be an elementary abelian 2-grou p of collineations on 7r,

with IF(A)~_ (P, Q, R). Let C be plane-maximal in A. Then one of the followin9 holds: (i) DZ(A)is a 2r-root subplane and C = A. (ii) g:(A) = (P, Q, R), and I A : C J = 4. In this case, A induces A(P, Q, R) on ~:(C). (iii) ~:(A) is the configuration of fix-elements of a homology on g:(C) centered at a vertex o f ( P , Q, R ) with axis the opposite side. Proof If C = A, then we are in case (i). Suppose C # A. Let a s AkC. Since C is plane-maximal in the abelian group A, the involution ~r must act as a homology in 0:(C). Since (P, Q, R ) c Dz(~r),the homology ~r must be centered at one of P, Q, or R with axis the opposite side. If all the elements of A\C are centered at the same point, then we are in case (iii). If cr and ~ are elements of A\C centered at distinct vertices of (P, Q, R), then by Result IV any element of A\C acts either as a, z, or ar on U:(C). Thus A induces A(P, Q, R) on 0:(C), and L e m m a 1 implies that [ A : C[ = 4. [] We shall have occasion to use the following special application of this lemma, whose proof is immediate. C O R O L L A R Y . Let A be as in Lemma 2, and suppose [A [ = 4. If ~(A) = (P, Q, R), then A = A(P, Q, R).

3. T H E E V E N - O R D E R

CASE

Assume now that o(zr) = n is even, not a fourth power, and that G is a group of collineations, transitive but not regular on the flags of zr. It suffices to produce in G a perspectivity. By a result of Ott [5, Satz 2], G has even order. Let Z be a maximal planar 2-group in G, and let % be the subplane I:(Z). If zco = ~, then by Result IV G has a perspectivity, so we m a y assume that o(~ro) = x/-n. Let Go be those collineations in G that stabilize ~o. We claim first that every flag in zro is fixed by a nontrivial elation in G~°. Indeed, let (P, l) be a flag in 7to, and let S be a 2-Sylow subgroup of Gp. containing Z. By a result of Ott [-5, Satz 4], S ¢ Z. Therefore, by the maximality of Z, there is a collineation in S that induces a nontrivial elation on 7ro and fixes (P, l). (Any element from S\Z that normalizes Z and whose square lies in Z will do.)

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N o w let C = IZ(Go) c~ no. Since n is not a fourth power, o(no) # 4. Thus, by a result of Piper [2, 4.3.23], if C = ~ , then no is Desarguesian, and G~° and contains the little projective group, while if C # ~ , then either C is a single point P and G~° is (P, P)-transitive, or C is a single line I and G~° is (l, /)-transitive. Thus, whether C is e m p t y or not, there exists in n o an axis l and in G O a s u b g r o u p T that induces the full translation group on n o relative to I. (Dualize if necessary.) We now show that T contains a nontrivial elation on n. Indeed, let K be the kernel of the action of T on n o. N o w T / K = T ~° is an elementary abelian 2-group of order n, and since T permutes the n - x / n points on/~n o a m o n g themselves, one of these, call it Q, is left fixed by an element fl of

T\/¢ This fl is the desired elation: since f12 E K, it fixes n o and Q, and therefore all of n pointwise. Thus f12 = 1. But since fl fixes m o r e than ~ + 1 points on l, it must in fact be an elation on n, and since not in K, is not trivial.

4. T H E O D D - O R D E R

CASE

Assume now that o(n) = n is odd, and that G is transitive on the flags of n. Let (P, l) be a flag in n, and let TO be a 2-Sylow subgroup of Ge. ~. We shall m a k e frequent use of the following lemma: L E M M A 3. Suppose n is an odd square. Let T be a 2-Sylow subgroup of G that contains T o. Then I T: To[ = 2, and for each tr ~ T\To, we have: (i) r = (a, To) , (ii) cr normalizes T o , (iii) a 2 ~ To. Proof. Since n is an odd square, 2 is the largest power of 2 dividing n + 1. Thus, by Result II, I T : To ] = 2, since n z + n + 1 is odd. The result follows. [] L E M M A 4. Let T be a 2-Sylow subgroup of G. Then Y(T) is a nonincident point-line pair. Proof. Since n 2 -t- n d- 1 is odd, there exists a tr ~ G with T" < Ge and a z ~ G with T ~ < Gz. Thus p , - 1 and l~-1 are in 0:(T). Since each flagstabilizer has even index in G, the configuration ~:(T) can contain no flag, and the result follows. [] LEMMA angle.

5. I f n is a square, then ~-(To)= ( P , Q, R ) , a nondegenerate tri-

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P r o o f We show first that no three points of I:(To) are collinear. Let Q be a point in IZ(To) and let TQ be a 2-Sylow subgroup of GQ with TO < TQ. Let a ~ T o \ T o . Since a normalizes To, it permutes the elements of I:(To). Since (a, To) = TQ, the only fix-elements of a in IZ(To) are Q and a single line not incident with Q. Suppose R ~ S are points of l:(To) each distinct from Q. Now a must act as an involution on OZ(To), since a z ~ To. Thus, both R R ~ and S S ~ must be the unique line of F(T0) fixed by a. This makes it impossible for R and S to be collinear with Q. Now let TF be a 2-Sylow subgroup of G e with TO < T e. Let IZ(Te) be the nonincident pair {P, m}. Let a ~ T p \ To . Let Q -- lm and R = QL Then I:(To) contains the nondegenerate triangle (P, Q, R). If it contained an additional point, then by the remarks above it would be a subplane of order 1, which is absurd. []

We are now ready to complete the proof of the main theorem. For the rest of this section, assume that n is odd, not a fourth power. By Result III, it suffices to show that G has a nontrivial homology. Thus, suppose that G has only the trivial homology. Result IV implies that n must then be a square. It will be convenient to let IZ(To)= (P1, P2, P3). For each i = 1, 2, 3, let Tei be a 2-Sylow subgroup of Gei with TO < Tel, and let ai ~ Tpi\To. Then a~ fixes P~ and interchanges Pj and Pk. Let Ao be the group generated by the central involutions in To. We shall reach a contradiction via the following sequence of steps: (1) I:(Ao) is a subplane o f order .x/-n. Since A o is characteristic in To, each ai normalizes Ao and so permutes the elements of I:(Ao). Since ai interchanges Pj and Pk, Lemma 2 implies that l:(Ao) must either be a subplane

or the nondegenerate triangle (P1, P2, P3). Suppose the latter. Since a 2 ~ To and Ao is central in To, the element ~rl acts via conjugation on Ao as an involutory automorphism. The al-composition factors of Ao are thus of order 2. Let B be a minimal al-invariant subgroup of Ao with t=(/3) = (Pt, P2, P3)- Since B contains no nontrivial homologies, the corollary to Lemma 2 implies that I B[ > 4. Thus, B has two distinct al-invariant subgroups B~ and B2, each of index 2 in B. We consider now the configuration tC(Bi). Since IZ(B) is a triangle, I:(Bi) cannot be a subplane and, by the minimality of B, cannot be (P1, P 2 , P3). Thus, IZ(Bi)must be the fix-elements of a homology on a subplane rcl. Since n is not a fourth power, the order of z:~ is x//-n. Since B~ is a~-invariant, this homology has center PI and axis P2 P3.

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Now let D = B1 c~ Bz, and consider fl:(D). It cannot be a triangle because of the minimality of B. If it were the fix-elements of a homology, then, since UZ(B3 _ ~:(D), we would have gZ(B1) = gz(B2). If ~:(D) were a subplane, then D would be plane-maximal in B~, so gx = ~:(D) = rc2 , and we would again have ~:(B~) = ~:(B2). Thus, in either case, we would have D:(B0 = IF(B1) c~ 0:(B2) = U:((Bx, B2)) = ~(B) = ( P l , P2, P3), which is a contradiction. Thus, g:(Ao) must be a subplane and clearly has order xfln. Let rco = 0:(Ao). Since To normalizes Ao, it induces a group of collineations on rco . Let Mo be the kernel of the action of To on rco . (2) TO contains a subgroup acting on rco as &(P1, P2, P3). Indeed, let c~ ~ To\M o with ~z e Mo. Then e acts as an involution on rco and so must be a (P~, Pj Pk)-homology, since n is not a fourth power. Let/~ = ~'J. Then fl acts as a (Pk, PjP~)-homology on rco, and the group ( e , / 3 ) clearly induces A(Vi, P2, P3) on fro. In light of the above fact, L e m m a 1 implies that the factor-group To/M o contains exactly three involutions. Furthermore, in To itself, each involution must act on 7zo either trivially (and so be in Mo) or as a (P~, PjPk)" homology. Let Cel be the (possibly empty) set of those involutions of To acting as homologies on rro centered at Pi. Since Cpi = C~ekj,each of the sets Ce~ must be of the same cardinality, and since ~ ~ Ce~ implies Cp~ ~_ c~Mo , this cardinality cannot exceed that of M o . (3) I f ] C e i l = [Mo], then Ce~ centralizes Cp~. Indeed, let ~ ~ Ces and f l e Cek. By Result VI, Ce~ ~ aflMo. Since these two sets have the same cardinality, they are actually equal. Thus, aft ~ Cp~ and so is an involution. N o w let (S, m) be a flag in n o . Let T1 be a 2-Sylow subgroup of Gs, m with Mo < T1. Since G is flag-transitive, the facts established above at (P, l) hold equally well at (S, m). Thus, if A1, rq, and M1 play roles in T~ analogous to those played by Ao, r~o, and Mo in To, we know, for example, that ]Mot = [ M1 [ and that T1/M ~ has exactly three involutions. Furthermore, if gZ(Tl) = (Qi, Q2, Q3), then Ti contains a group acting as A(Q1, Q2, Q3) on hi. We can, in fact, show even more: (4) T1 contains a group acting on n o as A(Q 1, Qz, Q3). Indeed, if M o n Ma ~ 1, then n o = n~, and we may use the group guaranteed by fact (2) above. Thus, we may assume that Mo c~ M~ = 1. In this case, Mo is isomorphic with a subgroup of T~/M~ and so has exactly one or exactly three involutions.

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225

In view of L e m m a 2, it suffices to show that T: contains an elementary abelian 2-group B that preserves Zo and has (Q1, Q2, Q3) as its fix-element configuration in z~o . If M o has three involutions, then these must generate a group that acts on n 1 as A(Q:, Q2, Q3). Thus, in this case, Z~o ~ 7:1 = UZ(Mo) n ~1 = . Now let B = A~. Then B centralizes A o and so preserves ~o. Furthermore, IF(B) ~ zco = z~1 ~ rco = . If M 0 has just one involution, then so does M r Let % and cq be the involutions in M 0 and M:, respectively. Let CQI play the role in T~ analogous to that played by Cp~ in T0. Since c% is an involution in T:\M:, we may assume it to be in C a r Let p E C~2. We claim that p centralizes % . Indeed, suppose this were not the case. Then the centralizer of p in Mo is trivial, since So is the only involution in Mo. This implies that [pMO[ = I Mo[. But since Mo fixes (Q1, Q2, Q3) pointwise, we have pMO : Co2, which implies that [ Ce, I = I Mol = [Mlf. Fact (3) above now implies that p centralizes % . N o w let B = ( e : , p). Then B preserves z o . Furthermore, (p, So) acts on 7:1 as A(Q I, Q2, Q3). Thus, ~(B) ~ too = I:(~1, p ) ~ IF(So) = ~(cq, p, So) = O:(~)

c~ IF(p,

%)

= 7:: c: IF = ( 0 t , Qz, Q3). N o w the group ( a : , a2, To) preserves ~o and leaves no element fixed. Hence, every point of no is the center of a nontrivial homology. Therefore, by a theorem of Piper I-2, 4.3.30], we have: (5) 7ro is Desarguesian, and G contains a group that acts on 7:0 as the little projective group. Thus, G contains subgroups H and K with K <~ H such that H / K is isomorphic with PSLa(q), where q = x//'n. Let 2 r be the largest power of 2 dividing q - 1, and let p be a primitive T - r o o t of unity in GF(q), the field with q elements. The element of PSL3(q) corresponding to the matrix

(:-°il P 1

0

1

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J.B. FINK

clearly has order 2 "+ 1. F r o m this, it follows that: (6) G has an element ~ of order 2 r+l. N o w let a be the involution in (~). Since we are assuming G to be homology-free, I:(~) must be a subplane of order q. N o w z stabilizes I:(a) and evidently permutes the points of rc not in I:(o). The n u m b e r of such points is N---(q4+q2+

1)-(q2+q+

1)=q(q2+q+

1)(q--l).

Since ~ has no fixpoints outside I:(~), its order must divide N, which, since q is odd, implies that 2 r + 1 divides q - 1, a contradiction. REFERENCES 1. Baer, R., 'Polarities in Finite Projective Planes', Bull. Amer. Math. Soc. 52 (1946), 77-93. 2. Dembowski, P., Finite Geometries, Springer-Verlag, New York, 1968. 3. Ostrom, T. G., 'Double Transitivity in Finite Projective Planes', Canad. J. Math. 8 (1956), 563-567. 4. Ostrom, T. G. and Wagner, A., 'On Projective and Affine Planes with Transitive Collineation Groups', Math. Z. 71 (1959), 186-199. 5. Ott, U., 'Fahnentransitive Ebenen gerader Ordnung', Archly. Math. 28 (1977), 661-668. 6. Wagner, A., 'On Perspectivities of Finite Projective Planes', Math. Z. 71 (1959), 113-123. Author's address:

J. B. Fink, D e p a r t m e n t of Mathematics, K a l a m a z o o College, Kalamazoo,

M I 49007, U.S.A. (Received October 5, 1980; revised version, March 21, 1984)