ALAN RAHILLY

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ABSTRACT. The existence of Baer collineations in a projective plane is related to the existence of desargues-like configurations. The plane of order four is characterized as the only finite plane that possesses a Baer subplane partition into tangentially transitive Baer subplanes which is preserved by each of the tangentially transitive groups. It is shown that a finite projective plane has either no or one tangentially transitive Baer subplane or is partially transitive of Hughes type (4, m), (5, m) or (6. m) for some m. The Lenz Barlotti classes which contain a finite plane which is not a translation plane nor its dual and which possesses a tangentially transitive Baer subplane are shown to be classes 1.1 and II.1.

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

In the theory of projective planes the concept of (P, /)-transitivity has proved to be most useful. By relating the amount of (P, /)-transitivity in a projective plane to the structure of some coordinate systems of the plane, it has been possible, by algebraic means, to decide questions of existence or non-existence of certain types of plane. For example, one can conclude from the nonexistence of finite alternative division rings which are not fields, that there are no finite non-pappian Moufang planes or, on the other side of the coin, given a quasified Q there exists an associated translation plane with Q as coordinate system. In fact, the situation in the finite case revolves around that of translation planes for almost every known finite projective plane is either a translation plane, a dual translation plane or derived from a dual translation plane, the planes in the derivation/dualization chain based on a Hughes plane being conspicuous exceptions. Not surprisingly, then, the concept of (P, /)-transitivity has supplied us with a most important means of classifying projective planes. In consequence the Lenz-Barlotti classification, based upon (P, /)-transitivity, lies at the centre of our endeavours to find what sort of projective planes there are. However, there is a need for the exploitation of the potential of other concepts. This paper concerns itself with the concept of 'tangential transitivity relative to a Baer subplane' and its central aim is to throw light on what sort of planes there are that possess a tangentially transitive Baer subplane. Two remarks need be made here. Firstly, our main aim is to consider the case of finite tangentially transitive planes and we consider only the finite case from Section 4 onwards. Secondly, we do not consider tangential transitivity relative to non-Baer subplanes and, in fact, we adopt a definition of'tangential transitivity' that explicitly excludes other than Baer subplanes. Thus our definition is somewhat different to that used by Jha [7]. Tangential transitivity (in our sense) bears a strong resemblance to (P, l)Geometriae Dedieata 13 (1982) 173-194, 0046-5755/82/0132-0173503.30. Copyright ~ 1982 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.

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transitivity. In Section 2 we draw out this analogy and pose some questions which motivate the work contained in the rest of the paper. In Section 3 we relate tangential transitivity to geometric structure analogously to the way that the amount of (P, /)-transitivity of a plane can be related to the amount that the plane is (P, l)-desarguesian. Many arguments which apply to the (P, /)-transitivity case carry over mutatis mutandis to the case of tangential transitivity relative to a Baer subplane by virtue of the fact that, in each case, the group under consideration fixes pointwise what Hughes and Piper [6] call a 'closed Baer subset', and so a lengthy discussion is most often unnecessary. In Section 3 we also relate tangential transitivity to some ideas first considered by T. G. Room in his paper [16]. In Section 4 we consider finite planes that possess a Baer subplane partition into tangentially transitive Baer subplanes and establish the following characterization of PG (2, 4): rt is a projective plane of order m2 which possesses a Baer subplane partition into tangentially transitive Baer subplanes such that the partition is preserved by each of the tangentially transitive groups if and only if re is PG(2, 4). In Section 5 we use this result to obtain the following classification theorem for finite tangentially transitive planes: if rc is a finite tangentially transitive plane of order m 2 relative to at least one Baer subplane then, either (a) the tangentially transitive Baer subplane is unique, or (b) ~ is a partially transitive plane of Hughes type (4, m + 1), (5, m) or (6, x/m). In Section 6 we consider the possible Lenz-Barlotti classes with finite tangentially transitive planes in them, and we show that, if rc is a finite tangentially transitive plane relative to a Baer subplane, which is not a translation plane nor its dual, then rc is in Lenz-Barlotti class 1.1 or II.1. 2.

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Throughout this paper n will denote a projective plane, ~ the set of points of rc and ~ the set of lines. We shall denote ~ u ~ by do and assume that ~ 2a is empty. Members of do will be called elements of n and a subset of dowill be called a configuration ofzt. A configuration ~f will be called closed if the element of rc jointly incident with any pair of distinct elements of the same type in c6 also belongs to ~. If ~ is a closed configuration then any element of do\~ is called a tangent element relative to ~, if it is incident with an element of c~. An element of do\~ which is incident with no element of is called an ordinary element relative to ~. Clearly a tangent element relative to a closed configuration c~ is incident with exactly one element of ~f. A configuration ~ is called a Baer subset if every element of n is incident with

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an element of ~. A Baer subset that is also a closed configuration is called a closed Baer subset. A Baer subset that is also a proper subplane of n is called a Baer subplane. A collineation which fixes a Baer subplane elementwise is called a Baer collineation. Throughout this paper we shall adopt the notation that '[P]' stands for the set of lines incident with the point P and '[l]' stands for the set of points incident with the line 1. By a simple enough argument (see [6], p. 92) it is possible to show that a closed Baer subset ~( :p g) is one of three types: (a) ~ n ~ = {Qo} u [Ko], 5 ~ n ~ = {K0} u [Q0], where Qo,Ko is a nonincident point/line pair, (b) ~ n ~ = [Ko], ~ n ~ = [Qo], where Qo, Ko is a flag of ~z, (c) ~ is a Baer subplane of re. It is well-known that the closed Baer subsets (a), (b) and (c) are 'maximal' proper closed configurations in the sense that, if c~ is a closed configuration properly containing such a closed Baer subset, then cg = g (see [6], p. 93). It follows that, if the fixed closed configuration of a collineation properly contains a closed Baer subset of type (a) or (b) or (c), then the collineation is the identity collineation. From now on we shall refer to closed Baer subsets of types (a), (b) and (c) as proper closed Baer subsets. We shall denote the group of all collineations fixing elementwise a proper closed Baer subset ~ by G(~). The complete fixed closed configuration of such a collineation must be either ~ or g, or alternatively, a non-identity collineation fixing ~ elementwise fixes all and only the elements of ~. We shall commence with a proposition concerning proper closed Baer subsets which is valid for certain other closed configurations as well. However, we are concerned primarily to emphasise the analogy between types (a) and (b) and type (c). P R O P O S I T I O N 1. Suppose JJ is a proper closed Baer subset of a projective plane n and that the collineation group G(~) is transitive on the set (assumed non-empty) of tangents relative to ~ incident with an element X of ~. Then G(~) is transitive on the set of tangents incident with any element Y of ~ for which the set of tangents is non-empty. Remark. The clauses about sets of tangent elements being non-empty are included in the statement of Proposition 1 since, if ~ is a closed Baer subset of type (a) or (b), then the point Q0, and the line Ko, of ~ are incident with no tangents relative to ~. The proof of Proposition 1 is straightforward and is left to the reader.

DEFINITION. Let ~ be a proper closed Baer subset of a projective plane n. Then n is said to be pointwise tangentially transitive relative to ~, if G(~)

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is transitive on the set of tangent points relative to ~ incident with some line of N' whose set of incident tangent points is non-empty. A dual definition can be framed for the term 'linewise tangentially transitive relative to ~'. Remarks. (a) In view of Proposition 1, pointwise tangential transitivity relative to ~ is equivalent to G(N) being transitive on the set of tangent points incident with any particular line of ~ whose set of incident tangent points is non-empty, a dual results holds for linewise tangential transitivityand also pointwise tangential transitivity relative to N is equivalent to linewise tangential transitivity relative to ~. (b) If N is a proper closed Baer subset which is not a Baer subplane and 7r is pointwise tangentially transitive relative to ~, then ~ is (Qo, Ko)-transitive. where Q0 and Ko are, respectively, the point and line constituting .~. (c) If N' is a proper closed Baer subset of ~ which is not a Baer subplane, then Proposition 1 implies that ~z is (Qo, Ko)-transitive if and only if rc is (Ko, Qo)-transitive, where Q0 and Ko are, respectively, the point and line constituting ~. Thus zr is (Qo, Ko)-transitive for appropriate Qo and Ko, if zr is linewise tangentially transitive with respect to N'. (d) If N is a Baer subplane of re, then zc is pointwise tangentially transitive with respect to ~ if and only if zr is linewise tangentially transitive with respect toN.

DEFINITION. Let N be a proper closed Baer subset of a projective plane ~. We shall say that ~ is tangentially transitive with respect to ~ if ~ is pointwise (linewise) transitive with respect to ~. In what follows we shall abbreviate 'tangentially transitive' to 'tt'. Remarks. (a) Proposition 1 also applies when 'proper closed Baer subset' is replaced by 'proper subplane'. If this substitution is made is the previous definition, then we have, essentially, the definition of tangential transitivity of Jha [7]. In this case the clause 'whose set of incident tangent elements is non-empty' is, of course, superfluous. (b) Clearly, the projective plane ~ is tt relative to a Baer subplane ~o if and only if the dual plane zra of rc is tt relative to rc~, the subplane of zca corresponding to zco.

DEFINITION. I f N is a Baer subplane and zcis tt relative to ~) then we shall say that B is a t t Baer subplane of ~. P R O P O S I T I O N 2. Suppose zr is tt relative to the proper closed Baer subset ~. Then G(N) is regular on the set of tangent elements relative to ~ incident with any particular element of ~ whose set of incident tangent elements is non-empty. Proof. Let X be a tangent element relative to ~' and Gx be the stabilizer

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of X in G(~). Then the closed configuration fixed by Gx properly contains and so is g. So we have that G~ is trivial and the result follows, Our treatment of tangential transitivity relative to a closed Baer subset has thus far been somewhat cumbersome due to the problems that arise out of treating the three types of proper closed Baer subsets together. However, our treatment draws out the clear analogy between tangential transitivity relative to a Baer subplane and tangential transitivity relative to a proper closed Baer subset of the other two types, which is, of course, (Qo, Ko)transitivity, and this motivates the following sequence of questions. QUESTION 1. Can we relate pointwise and linewise tangential transitivity relative to a Baer subplane to the internal geometric structure of the plane in an analogous way to the way that (P, /)-transitivity is related to the existence of desarguesian and dual desarguesian configurations ? QUESTION 2. It is well-known that a certain amount of (P, /)-transitivity in a plane often implies further (P, /)-transitivity. Can we get some interesting analogous results for tangential transitivity relative to a Baer subplane? QUESTION 3. Can we use theorems on tangential transitivity relative to a Baer subplane to obtain a reasonable classification of planes that contain tt Baer subplanes in the spirit of Lenz and Barlotti ? QUESTION 4. What is the 'maximum amount' of tangential transitivity relative to Baer subplanes that a plane can have and which planes possess this amount? Remark. To answer Question 4 it is necessary to decide what the term 'maximum amount' means. For instance, PG(2, 4) possesses many Baer subplanes and is tt relative to all of them. So the maximum amount might be considered to be tangential transitivity relative to all Baer subplanes. This, however, has the drawback that any plane without a t t Baer subplane (for example, Pappian planes of order greater than four, finite planes of non-square order, etc.) has the maximum amount vacuously. This is in contradistinction to the case of (P, /)-transitivity for which the desarguesian planes have the maximum amount.

QUESTION 5. The collineation groups of a plane will permute tt Baer subplanes amongst themselves. What is the nature of the constituents of collineation groups (not necessarily the full collineation group of the plane) on the set of tt Baer subplanes ? Remark. In their paper [11] Praeger and Rahilly consider three classes of finite plane in which collineation groups act on a set of tt Baer subplanes respectively as PGL(2, pt), the group of linear substitutions on GF(p t) and PGL(3, 2), each in their natural representation.

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The material which follows should be viewed in the light of the above questions. We shall make a start with Question 1. 3. P S E U D O - D E S A R G U E S I A N

CONFIGURATIONS

D E F I N I T I O N . Let rc be a projective plane and n o a Baer subplane of re. Suppose l~, Iz, 13 a r e three distinct lines of n o and the point triples P1, P 2 , P 3 and Q1, Q2, Q3 define mutually disjoint triangles A 1 and A2, respectively, which are disjoint from n o and such that P1QI = 11, P2Q2 = 12 and PaQ3 = 13. Suppose further that R~ -- P1P2 ~ Q~Q2, RE = P2P3 (~ QEQ3 and R3 = PaP1 n Q3Q1 all belong to n o . Then we say that 11,12,13,A 1 ,Ae, R~, R2, R 3 form a pseudo-desarguesian configuration relative to ~z0 . Further, if there is a Baer subplane ~z0 of z~ such that, whenever 11,12,13, A 1 , A2 are chosen as above and R1, R 2 belong to n0, then R 3 also belongs to n0, then we say that ~zis pseudo-desarguesian relative to n o . If we define 'dual pseudo-desarguesian configuration' in an obvious manner, then a pseudo-desarguesian configuration with respect to a Baer subplane ~ is also a dual pseudo-desarguesian configuration with respect to 8 , and vice versa. Remark. The lines ll, 12, 13 may or may not be concurrent and the points R3 may or may not be collinear. If la, 12, la are concurrent and R ~ , R 2 , R3 are collinear then the pseudo-desarguesian configuration is then a desarguesian configuration. This happens, for instance, in a desarguesian plane of square order. Simply co-ordinatize ~ by the field F and 7zo by the square-root order subfield Fo and choose P1 = (t, 0), P2 = (t, t), P3 = (0, t) and Q1 = (S, 0), Q2 ---- (S, S), Q3 = (0, s), where s, t~Fo, s =p t. R1, R2,

We are now in a position to establish three theorems, each of which has a straightforward analogue in terms of (P, /)-transitivity, desarguesian configurations and dual desarguesian configurations.. T H E O R E M 1. A finite projective plane rc possesses a pseudo-desarguesian configuration if and only if it possesses a Baer subplane. Proof. One half of this is trivial, so suppose rc possesses a Baer subplane rc0 . Let 11,12, 13 be three distinct lines of re0 and A and B be two distinct points of rCo neither of which lies on 11,12 o r 13 . There are such A and B if m 2 ----= Ire [ > 4 and, if[~ [ = 4, then rc is PG(2, 4) and pseudo-desarguesian configurations exist in ~. Consider an arbitrary tangent point Xa relative to % on 11 and let X 2 = A X t c~ 12 and X 3 = B X 2 ( 3 13 . We define a function e from the set of tangents incident with 11 into ( ~ c~ no)\ ([/1] tJ [13] u {A, B } ) b y e(X1) = X a X a c~ no. But the domain of c~has exactly m 2 - - m points and the range of e at most m 2 - - m - - 2 points. Therefore, there are two triangles with vertices X'~, X~, X ; and X~, X 2 , X3 constructed H

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as above such that ~(X]) = ~(X~). But then these triangles yield a pseudodesarguesian configuration relative to n o. COROLLARY: If the f n i t e plane n possesses a Baer subplane % then n possesses pseudo-desarguesian configurations relative to no. The proof of Theorem 1 is an obvious modification of an argument due to Ostrom [9] which establishes the existence of desarguesian configurations in any finite projective plane. Let rc be a projective plane, no a Baer subplane, I a line of no and A and A 1 two distinct tangent points on I. We can define a function fl on the set of points of the affine plane n(1) obtained from n by deleting l, as follows: Denote the affine subplane of n(1) corresponding to n o by no(l). Then fl(P) = P i f P e ~ ~ no(l) and i f P 6 ~ c~%(0, then fl(P)= A 1 M ~ [ where M is no(1)~AP and l'is the unique line of no(l) containing P. It is not difficult to show that fl is a permutation of the points of n(l). Should fl induce a collineation of n(l) then this collineation will be extendible to a (Baer) collineation of n fixing Zo elementwise and which takes A to A 1The following lemma establishes a necessary and sufficient condition for this to happen. LEMMA. The permutation fi on the set ~ ~ n(l) induces a eollineation on n(1) if and only if the triangles A, B, C and AI , B1, C1, where Ba = fi(B), C 1 = fl(C), form a pseudo-desarguesian configuration for all points B, C of n(1)\no(l) such that the affine line l* containing B and C is not a line of no(l ) and A is not incident with the projective counterpart l' off*. Proof. Suppose/3 induces a collineation of n(l) and thus of n. Consider two points B, C of n(1)\rco(l) such that l* = BC is not a line of no(1) and A4l'. Clearly, the triangle A1, B1, C1 is the image of A, B, C under the induced collineation in It. Now l*qrCo(l) and so l' contains a unique point M' of ~zo. Also M' is on the image of l' which contains B1 and C 1 . It follows that the required triangles form a pseudo-desarguesian configuration. Suppose the triangles form a pseudo-desarguesian configuration. Let n* be a line of n(1). If n* contains two points of no(l) then clearly/3 permutes the points of n* amongst themselves. Suppose n* contains at most one point of no(l). The projective counterpart n' of n* contains a unique point N of no. Let P and Q be distinct affine points ofn' (distinct from N, if N~no(l)). If A, P, Q are collinear then A1, fl(P), ~(Q), N are collinear. If A, P, Q are not collinear then, by hypothesis, N,/3(P),/~(Q) are collinear. This establishes that /~induces a collineation on n(1). T H E O R E M 2. Let n be a projective plane, no a Baer subplane, I a line of reo and A, A1 two tangent points on I relative to n o. Then, rcpossesses a collineation

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fixing ~zo pointwise and taking A to A 1 if and only if the triangles A, B, C and A1, B1 = fl(B), C1 -- fl(C) form a pseudo-desarguesian configuration for all B, C belonging to ~z(l)\~zo(l) such that the line l' of rr containing B and C is not a line of~o and such that A is not on l'. Proof. This follows immediately from the preceding lemma. T H E O R E M 3. Let ~ be a projective plane and ~o a Baer subplane of ~z. Then, 7r is tt relative to ~zo if and only if lr is pseudo-desarguesian relative to ~zo. Proof. An obvious application of Theorem 2 suffices. Room [16] has formulated two axioms which he uses to characterize the odd order Hall planes. The condition that rc is pseudo-desarguesian with respect to rCo may be considered to be a variation on the first part of Room's second axiom. In [13] Room's Axiom 1 and the first part of his Axiom 2 are used to characterize finite generalized Hall planes. 4.

BAER

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Let rc be a projective plane of order m 2 and rco, ~1 . . . . . rcm2_mbe m 2 - - m + 1 mutually point- and line-disjoint Baer subplanes of ~. Such a collection of Baer subplanes is called a Baer subplane partition of re. In this section we wish to prove a theorem characterizing PG(2, 4). In order to do this we shall need the following result of Shult [17]: Suppose G is a doubly transitive permutation group on the set f~ and that the stabilizer G~ of an element ~ ~ f~ has a normal s u b g r o u p / ~ of even order which is regular on f~\{e}. Then, either (a) L ~ G ~< S, where L and S are the group of linear transformations and the group of semi-linear transformations, respectively, of a nearfield N F (q) of order q -- ]~ ], or (b) N ~ G ~< Aut N, and (i) q = [f~l = 1 + r, and N = PSL(2, r), or (ii) q - - [ f l ] = 1 + r z, and N = Sz(r), or (iii) q -- ]f~] = 1 + r 3, and N = U(3, r), where r = 2 n, for some n. Before proceeding to Theorem 6, the following lemma needs to be proved. LEMMA. If Tz is a projective plane of order m 2 which possesses a Baer subplane partition into tt Baer subplanes, then the length of each point orbit of the group generated by the tt groups of Baer collineations is at least ( q - 1)(q - 2 ) + 1, where q = m 2 - m + 1. Proof. Consider an arbitrary point P of zc and let rcl be the Baer subplane containing P in the Baer subplane partition. There are q - 1 = m 2 - m

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tangent lines with respect to ~ though P and each of the other q - 1 = m 2 - m Baer subplanes in the partition has one these as a line. The orbit of P under the tt group which fixes one of these Baer subplanes is of length q - 1. It follows that the length of the orbit containing P is at least (q - 1)(q - 2) + 1. T H E O R E M 6. 7r is a projective plane o f order m2(rn >~ 2) which has a Baer subplane partition consistin 9 of tangentially transitive Baer subplanes rex, rcz .... , rcq, where q = m 2 - m + 1, such that each o f the tt 9roups R~, R2 . . . . . Rq preserves the Baer subplane partition if and only if rc is PG(2, 4). Proof, Let C21 be a cyclic Singer group acting on PG(2, 4) and C3 and

C7 be the subgroups of CE~ of orders 3 and 7, respectively. Each point orbit of C7 is the point set of a Baer subplane of PG(2, 4) and the lines of such a subplane form a line orbit of C7. Consequently, these Baer subplanes constitute a Baer subplane partition of PG(2, 4). Furthermore, each of the Baer subplanes in this Baer subplane partition is tt, the tt groups being of order 2. Such a tt group normalizes C7 and so preserves the Baer subplane partition, and also normalizes C3, and so, in consequence, generates with C3 a group isomorphic to the symmetric group $3 on 3 letters. (Notice that the estimate of the orbit length in the preceding lemma cannot be increased for, if ~ is PG(2, 4), that is, m = 2, q = 3, then the orbit length under the group $3 generated by the tt groups is (q - 1)(q - 2) + 1 = 3.) Now for the converse result. Suppose ~ is a plane of order m 2 possessing a Baer subplane partition = {rc~[i = 1, 2 .... , q}, where q = m 2 - m + 1, and each of the subplanes rr~is a tt Baer subplane such that the corresponding group Ri of Baer collineations preserves the partition. Let G be the group generated by the R~ and G~ the stabilizer of ~z~in G. Now G permutes the elements of f~ amongst themselves. Let q5: G --+ G be the natural homomorphism of G onto its action on fL K the kernel of ~b and U = qS(U) - U / U c~ K, for any subgroup U of G. Clearly (~ is doubly transitive on f~ and, since R~ ~ G~, we h a v e / ~ ~ G~. Now each line of ~ contains exactly one point of 7rj, where i :p j, and so Ri c~ Gj = 1, i ¢ j . Clearly, Ri c~K = 1 and so Ri-~/~i, and/~i is regular on f~\{~} for each i. But I/~[( - m 2 - m) is even and so we may apply the result of Shult above. In case (b), rn 2 - m is an even prime power, which is impossible unless m = 2, whence rr is PG(2, 4). (Notice that in this case G = $3 - PSL(2, 2).) In case (a), G is a group of semi-linear transformations on a nearfield NF(q) (possibly a field), whose elements are those of fl, of order q = m z m + 1. Now/~i is a subgroup of G of order q - 1 and G contains a normal subgroup L (the group of linear transformations on NF(q)) which contains an elementary abelian subgroup A of order q. The group generated by/~1 and ~] is the semi-direct product of/~l and ~] and contains q conjugate subgroups of/~ 1 which must be the/~, i = 1, 2 . . . . . q. But then we see that the/~i generate

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this group of order q(q - 1). But the group generated by the/~i is G which contains L. Clearly G -- L. Now G is the group-disjoint union of the/li and A. The pre-image A of A in G is a normal subgroup of G and is group-disjoint from each Ri. The group generated by A and R 1 is the semi-direct product of these groups and contains all the Ri. So G is the semi-direct product of A and any particular Ri. We shall now show that G is transitive on the set of points of n, if m/> 3. Each point orbit of G is at least of length (q - 1)(q - 2) + 1 and this is greater than ½q(q + 2m), that is, half the number of points of n, if m >f 5. If m = 3, then (q - 1)(q - 2) + 1 = 31 and so, if G is not transitive on the set of points of n, then there are two orbits, one, F 1, of length e and the other, F2, of length fl, say, where ~, fl/> 31 and ~ + fl = 91. N o w the point set of any of the subplanes of fl cannot be contained in one of the two point orbits of G, since G is transitive on f~. So each of the seven Baer subplanes nontrivially pointwise intersects each of F I and F 2 . Within each of these point orbits the intersections of the orbits with the Baer subplane point sets will be transitively permuted by G, and these intersection-point sets will equipartition their respective orbits. So 7 must be a divisor of a, and of ft. This means ~ = 35, fl = 56 or ~ = 42, fi = 49, assuming, without loss of generality, that e 133. A similar analysis to that of the case m = 3 shows that 13 must be a divisor of both e and ft. But a simple check of the possible values of e and fl shows this to be impossible. We conclude that, in the case m = 4, G is transitive on the point set of n. F r o m now on we suppose that m ~> 3. Our next task will be to show that each of the non-identity elements of of K fixes no point of re, ifm >f 3, that is, K acts semi-regularly on n n ~ . Let k e K and P be a fixed point ofk. N o w P e r h, for some i, and k restricted to r~ is a collineation of zh. So, by a result of Baer, k fixes a line I of 7h. But l contains exactly one point of each rcj, i ~ j . So k fixes each of the m E - m

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tangent points relative to rci on I. Using one of these points we can obtain another line l' fixed by k with at least m 2 - m points on it fixed by k. Ifm >/3 we are guaranteed that four of the points on I and l' fixed by k form a quadrangle. Consequently, the fixed points of k form a subplane of zr, which is obviously the point set of re, whence we conclude that k = 1. Consider a point P of zcwhich is in the Baer subplane rci.Then q(q - 1)IKI = 10l Igr = IGI =-IG,[IPGI= [Gp/eilleilq(q + 2m)= IGe/Ril(q - 1)(q + 2re)q, whence [K I = [G~,/R~J(N + 2m), But Igl ~ q + 2m, since K is semi-regular on the point set of each zcl. It follows that Ig l = q + 2m and so K acts regularly on the point set of each ~i. Also IG [ = q(q - 1)(q + 2m) and IAI = q(q + 2m). Note that, since ~] is regular on f2 and K is regular on the point set of each rtl, A is regular on the point set of re. Next, let H be a Sylow p-subgroup of A, where p is the prime divisor of q. Since q and q + 2m are relatively prime H c~ K = 1 and so A is the semidirect product of H and K. Also, IH! = q and, in fact, H(~- A / K ~- 7t) is elementary abelian. Further, H acts semi-regularly on the point set of zr with orbits of length q and induces a regular permutation group on f~. In consequence, each of the point orbits of H contains precisely one point from each Baer subplane .,ri. Clearly, the Sylow p-subgroups of A are also the Sylow p-subgroups of G and the number of such subgroups is odd, being a divisor of q(q + 2m), in fact, of q + 2m. Now each of the groups Ri is isomorphic to the multiplicative group of a nearfield NF(q) (which, of course, might be a field) and so contains a unique involution ui. The group U1 = ( U a ) induces (by conjugation) a permutation on the Sylow p-subgroups of A. Since these are odd in number Ul normalizes at least one such subgroup H. But then the group V generated by H and U1 is the semi-direct product of H and U1. Furthermore, V contains each of the involutions uj,j = 1, 2 . . . . . q, since H is transitive on f~, and is, in fact, the group generated by the uj, which is a normal subgroup of G. But the only Sylow p-subgroup in V is H and, in consequence, H is the unique Sylow p-subgroup of G (and of A and V) which means that H is a normal Sylow p-subgroup of G. But, then R1 normalizes H and so permutes the point orbits of H. However, such a point orbit of H contains exactly on point of rh. So R1 fixes each point orbit of H. It follows that each point orbit of H consists of precisely one point of zrl plus the tangent points relative to 7zi on a line of 7t1. Similarly, since R2 normalizes H, each point orbit of H consists of precisely one point of rc2 plus the tangent points relative to rc2 on a line of 1r2. But these compositions for the orbits of H are jointly impossible, if m ~> 3. This contradiction eliminates case (a) with m i> 3 and the theorem is established.

Remarks. (i) Notice that a similar situation to that described in considering case (a) in the proof of Theorem 6 actually occurs in PG(2, 4). There are 3

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Baer subplanes rc~ each with a tt group Ri of order 2. There is a group A of order 21 with group-disjoint normal subgroups H and K of orders 3 and 7, respectively. The group G generated by A and one R/is of order 42. The group generated by one R/ and /-/is of order 6 and is that generated by the R/ together. The point orbits of H consist of a point of rei plus the two tangent points relative to ~zi on a line of rh,for each i = 1, 2, 3. (ii) The group G = $3 - L on GF(3) in the PG(2, 4) case. 5. C L A S S I F I C A T I O N

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In this section we wish to prove a theorem which gives a reasonable classification of finite planes that are tt relative to a Baer subplane. For this we need the concept of a 'partially transitive plane'. D E F I N I T I O N . Let 7z be a finite projective plane, G a collineation group of re and zc' the fixed configuration of G. If G acts regularly on the set of ordinary points, and on the set of ordinary lines, relative to re', then ~ is said to be a partially transitive plane relative to G and r~'. We shall be particularly interested in the following three cases: (a) re' consists of m points (m/> 3)Qi, i = 1, 2 .... , m, on a line Ko, and a point Qo not on Ko, and the m + 1 lines Ko, QoQi, i = 1, 2 ..... m. Hughes [4] calls this type (4, m). (b) rt' consists of m + 1 points (m >t 2)Qi, i = 0, 1.... , m, on a line Ko, and m + 1 lines Ki, i = 0, 1. . . . . m, each through Q0. Hughes [4] calles this type (5, m). (c) re' consists of m E q- m d- 1 points Qi and lines Ki, i = O, 1, ..., m 2 q- m , which constitute a subplane of rc of order m. Hughes [4] calls this type (6, m). In each of the cases it is assumed that the sets of ordinary points and lines are non-empty. In what follows we adopt the following notation: Let A _~ ~ and H _~ ~'. Then, [A] denotes the set of points incident with at least one member of A, and [H] denotes the set of lines incident with at least one member of of H. Also, let Xi, i = 1, 2 .... , r, be a collection of configurations of the projective plane ft. We shall say that the X / f o r m a point cover of the subset Y of 5~ if [Y] _ u / ( ~ n X / ) , and a line cover of the subset Z of ~ if [Z] _ u / ( ~ n Xi). If these set inclusions are set equalities, we shall say that the X/form an exact point or line cover. In [4] Hughes shows that (a) if ~ is partially transitive of type (4, m), then [~z[ = (m - 1)2 and there are m tt Baer subplanes ~o,-.-, ~m-1 such that ~ / n ~j = n', i @ j , and the ~/form an exact point cover of {K/[i = 1, 2 . . . . . m} and an exact line cover o f { Q / I / = 1,2 . . . . . m},

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(b) if 7r is a partially transitive plane of type (5, m), then In I = m2 and there are m tt Baer subplanes no . . . . . n,,_ 1 such that ni n nj = n', i ~ j , and the ni form a point cover of K = {K~li = 1. . . . . m}, a line cover of Q = {Q~li = 1. . . . . m} and u~(rc, n ~ ) = [ K ] u Q and u ~ ( n i c ~ ) = [ Q ] u K , and (c) if rc is a partially transitive plane of type (6, m), then In l = m* and there a r e m 2 -F m + 1 tt Baer subplanes no, ..., nm2+,, such that n~ n nj = n', i @j, and the ni form an exact point cover of {Kil i = 0, 1,...,/In 2 + m} and an exact line cover of {Qi[i = 0, 1. . . . . m 2 + m}. T H E O R E M 7. Let n be a finite projective plane of order m 2 (m ~ 2) and suppose that n is tt relative to at least one Baer subplane. Then, (a) n has a unique tt Baer subplane, or (b) n is of Hughes type (4, m + 1), (5, m) or (6, x / ~ ) . Before proceeding to the proof we shall establish a lemma. LEMMA. Suppose n o and re1 are two tt Baer subplanes of a plane n of order m z. Suppose also that n ' = n o n nl is a closed Baer subset o f n o and na. Then (a) /fzr' is a closed Baer subset of type (a) (see p. 175), then 1r is of Hughes type (4, m + 1) relative to n', (b) /f zc' is a closed Baer subset of type (b), then n is of Hughes type (5, m) relative to n,, (c) /fzr' is a closed Baer subset of type (c), then n is of Hughes type (6, x/m) relative to re'. Proof. Let Ro and R1 be the tt groups on no and z~1 , respectively, and G the group generated by Ro and R1. If g~G fixes an ordinary point P relative to n' then, in the first two cases, g fixes m + 2 lines through one particular point of 7r', and, in the third case m + xf r o + 1 lines through P. Since g must fix a proper subplane of n we have that g = 1 and so G is semi-regular on the set of ordinary points relative to n'. A similar argument shows that G is semi-regular on the set of ordinary lines relative to n'. But IG I I> [Ro 12 = m2(m - 1)2 and since there are, respectively, (m + 1)m(m - 1) 2, m3(m - 1) and m(m - 1)(m 2 - x/m) ordinary elements of each type we see that G is regular on the sets of ordinary elements in each case, except possibly case (b) with m = 2, which is easily handled separately. Proof (of Theorem 7). Suppose there are at least two tt Baer subplanes. If all pairs of tt Baer subplanes are mutually point- and line-disjoint, then we readily have that the complete set of tt Baer subplanes form a Baer subplane partition which will be preserved by the tt groups associated with the Baer subplanes in the partition. By Theorem 6, n must be PG(2, 4), which is of Hughes types (4, 3) and (5, 2). Suppose there are two tt Baer subplanes which share at least one point. We show that n is of Hughes type (4, m + 1), (5, m) or (6, x / ~ ) . Before proceed-

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ing we note that the supposition that there are two tt Baer subplanes which share at least one line leads, using the a r g u m e n t to follow, to the conclusion that the dual of rc is of Hughes type (4, m + 1), (5, m) or (6, x/m), and hence rc is of one of these types. N o w consider all triples no, rq, l, where rco, ~za are tt Baer subplanes of n and l is a line of r~ such that ~ c~ n o ~ rq contains a point of I. Take a triple such that ] ~ c~ n0 c~ n~ c~ [/] [ is the maximal value # for re. We divide the p r o o f into a consideration of three cases: # = m + 1, # = m and 1 ~< # ~< m - 1. We, in fact, shall show that ~t = m + 1 leads to n being of Hughes type (4, m + 1) or (5, m), that # = m implies m = 2 and that, if 1 ~< # ~< m - 1, then m is square, kt = x f m + 1 and n is of Hughes type (6, x/m). Case 1/~ = m + 1. If no and ~a share a further point it will be unique. By applying part (a) of the preceding lemma we have that ~ is of Hughes type (4, m + 1) relative to n' = rco c~ n~. If rco and rq share no further points then, since they are Baer subplanes, they linewise share exactly a pencil of lines. Thus re' = n o n na is a Baer subset of ~zo and n~ of the type considered in part (b) of the preceding lemma and so we have that n is of Hughes type (5, m) relative to n'.

~gl)\[ll. Then no and the m 2 - m images of rc~ under the tt g r o u p of rco share m lines as well as I. But these lines can a c c o m m o d a t e at most m + 1 Baer subplanes without further intersection of subplanes. But m 2 - m + 1 > m + 1, if m > 2, and so we have m = 2. Suppose that [ ~ c~ rco c~ rh I = m. Since rco is a Baer subplane of r~ every point of n~ lies on a line of rt 0 and thig readily yields that there is a line l' of reo c~ n I other than 1. N o w there are m 2 - m images ofrc~ under the tt g r o u p on n 0 , and l' cannot a c c o m m o d a t e all of these without further subplane intersection. But then the a r g u m e n t of the preceding p a r a g r a p h m a y be applied. Case 2 # = m. Suppose that there is a point 0 e ( ~ C ~ n o

Case3 l~<#~2, then a line / f o r which I ~ o ~ n l ~ [ q l is maximal will be c o m m o n to ~Zo and re1. If # = 1 we show that no, ~z~ and l can be chosen so that N n r~o n nl = {P}, P e l , and l e 5 ° c~ no c~ rq. Suppose # = 1 and n~ and n¢ are two tt Baer subplanes such that N c~ ~ n rc~ = {P}. Suppose there is no line l such that P ~ l and l ~ Y c ~ n , n ~ z p . Then, two images ~o and n~ of np under the tt g r o u p of n~ must share P and a line t h r o u g h P, since the set of m 2 + 1 lines t h r o u g h P cannot be equipartitioned into sets o f m + 1 lines. N o w let us take, in any case where 1 ~< # ~ m - 1, a pair oftt Baer subplanes re0, nl and a line l c o m m o n to rco and n I such that I ~ o ~ n , ~ E t ] l = ~. Next let us introduce two points P, Q ~ ( ~ n n l c~ I / ] ) \ n o and a point P ' e (Pc~nl)\((~c~rc0)w [/]). N o w P P ' is not a line of rCo and so meets rCo in a

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unique point R. Further, P' is on a unique line l' of no and there is a Baer collineation a of no which takes P to Q and hence P' to Q' where Q ' = R Q n I ' . N o w a ( r q ) = re1 for otherwise na and a 0 h ) share more than # points on 1. Consequently, Q' = a(P') is a point of rq. So the line l' has two points of rq on it and thus l'E50 n no n ~a. Clearly, since P' was arbitrarily chosen, ~o n rh is a closed configuration of rq such that any point of n~ is on a line of it. If no c~ ~ is not a proper subplane of n~, then 5e n no n nl contains a complete pencil of lines of n~. If 5 ° n rco n rq has an extra line, then ~z0 n ~1 is a closed Baer subset of type (a) (see p. 175) of n I (and of no). If ~ n rco n g a has no further line then, since no and ~ are Baer subplanes, we have that ~o n ~a is a closed Baer subset of rq (and rCo) of type (b). But then, in either case, there is a line l* such that IN n n o n rq n l* I = m + 1 > ~t. It follows that n o n nl is a proper subplane of rq and, since each point of rq lies on a line of no n ~1, that m is square, and that no n na is a Baer subplane of nl (and of rCo) of order x/-m. By applying part (c) of the preceding lemma, we conclude that 7r is of Hughes type (6, xf-m).

6. LENZ--BARLOTTI CLASSES CONTAINING FINITE TT PLANES In [7] Jha has posed the question as to which Lenz-Barlotti classes contain finite tt planes. (We remind the reader that Jha's usage of 'tt' differs from our own.) In this section we answer this question in the case of planes which are tt relative to a Baer subplane. We shall make a start with the following two simple theorems. T H E O R E M 8. (i) Let rc be a partially transitive plane of Hughes type (4, m) relative to n'. Let n o .... , ~m- 1 be the m tt Baer subplanes which form an exact point cover of (5° n r~')\{Ko}, Ro . . . . . R,,_ 1 the respective tt groups and H the stabilizer of a point P c [Ko]\{Qi[i= 1. . . . . m} in the partially transitive group G. Then, H is a (Qo, Ko)-transitive group of homologies if and only if H is a normal subgroup of G. (ii) Let n be a partially transitive plane of Hughes type (5, m) relative to n'. Let n o .... , n,,_ ~ be the m tt Baer subplanes which form a point cover of (£/3 n rc')\{Ko}, Ro,... ,R i the respective tt groups and E the stabilizer of a point P c [Ko]\{Qi Ii = 0 , . . . , m} in the partially transitive group G. Then, E is a (Qo, Ko)'transitive group of elations if and only if E is a normal subgroup of G. Proof. (i) If H is a (Qo, Ko)-transitive group of homologies, then H ~ G since G fixes Qo and K o . Suppose H N G. N o w H stabilizes P and so l ul = I 1- 1 : m(m- 2), by Hughes [4], p. 655. We have thus only to show that each element h of H is a (Qo,Ko)-homology. But G is transitive on [Ko]\{Q~Ii= 1, ... ,m}

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(see Hughes [4] ) and so let x ~ G such that x(P) = Q, where Q is an arbitrary point in [Ko]\{Qi[i = 1. . . . . m}. Then x H x - 1 = H fixes Q. (ii) A similar argument suffices. T H E O R E M 9. Suppose ~ is a finite plane of order m 2 which is tt relative to a Baer subplane ~o and (Qo, Ko)-transitive, where Q o e ~ c~no and Koe~+c# c~ ~ro. (i) If Qo is not on Ko, then ~ is of Hughes type (4, m + 1) relative to rr', where ~ ~' = {Qo} w ( [Ko] ~ ~o) and ~e ~ ~' = {Ko} ~ ([Qo] ~ ~o). (ii) If Qo is on Ko, then 1r is of Hughes type (5, m) relative to ~', where

[Ko]

and

Proof (i) We shall show that the group G generated by the tt group Ro on 1ro and the (Qo, Ko)-transitive group H is regular on the set of ordinary points relative to n'. Clearly G is the semi-direct product of H and Ro. Suppose gmG fixes the ordinary point P. Now g = hr where h = H and r~Ro. We have that r(P) =- h- l(p) and so r fixes QoP which is a tangent line of ~o, whence r = 1, and then h = 1 as well. But [ G I equals the number of ordinary points relative to ~r'. A similar argument applies to the set of ordinary lines. (ii) A similar argument suffices. If n is a generalized Hall plane or an Ostrom-Rosati plane then 7r satisfies the hypotheses of Theorem 9 (ii). However, there are only three known planes which satisfy the hypotheses of Theorem 9 (i). They are PG(2, 4) and the nearfield planes of order 9. In fact, these are all, for we can prove: T H E O R E M 10. g is a finite plane of order (m - 1)Z(m >1 3) that is tt relative to a Baer subplane 7ro and which is (Qo, Ko)-transitive f°r a non-incident point/ line pair Qo, Ko °f ~o if and only if ~ is PG(2, 4) or a nearfield plane of order 9. Also the partially transitive group is GL(2, 2) or GL(2, 3). Proof. That PG(2, 4) and the nearfield planes of order 9 possess the required groups is shown in Hughes [4] (pp. 667-668), so suppose g is tt relative to ~o and (Qo, Ko)-transitive for a non-incident point/line pair Qo, Ko in 7r0 and let G be the partially transitive group which is the semi-direct product of the (Qo, Ko)-homology group H and the tt group Ro on 7to. Now the stabilizer Ho of ~o in H is of order m - 2 and Ro induces by conjugation an automorphism group of order (m - 1)(m - 2) which fixes Ho pointwise and which is regular on H\Ho. We proceed to show that G is either GL(2, 2) or GL(2, 3). Suppose m is divisible by two odd primes. Then since m - 2 is not divisible by either we have that H\Ho has elements of both these prime orders which is impossible. Suppose m -- 2Spr, where p is an odd prime and r >/1. If s > 1 then a Sylow 2-subgroup of H has order 2 s+ 1/> 8 and a Sylow 2-subgroup of Ho has order 2. It follows that H\Ho has elements of both odd and even orders, again impossible, whence s ~< 1, If m = 2p r then a Sylow 2-subgroup

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of H o has order 2 l+ 2, where 2 l[I pr _ 1, and a Sylow 2-subgroup of H has order 2 ~+2 and once again we have a contradiction. Clearly, m must be a prime power. Suppose m = pr where p is odd and r/> 1. Now p does not divide m - 2 and so all elements of H\Ho are of order p. Also H0 is normal in H. Now consider heI-I~Ho. This element induces an automorphism on Ho which must fix non-identity elements of Ho, if m > 3. Let ho be such an element in Ho. Then (hoh) p = hgh ~' = h g. But hoh~H\Ho and so ho has order p. It follows, by contradiction, that m = 3, Ho is trivial. H is cyclic of order 3, Ro is cyclic of order 2 and G is GL(2, 2). Suppose m = 2~, where s ~>2. Now a Sylow 2-subgroup of Ho is of order 2 and of H of order 2s+l and we have that each element of H\Ho is of order 2", where u ~> 1. If rio were not normal in H then there would be an involution e c H o and x ~ H \ H o such that e ' = x - l e x e H \ H o . Thus all elements of H\Ho would be involutions. But then (xe) 2 = x 2 = e 2 = 1 implies e = e', which is a contradiction. Therefore Ho ~ H. If u > 2 there are elements of order 2" and 2 u- 1 in H\Ho whence, by contradiction again, u ~< 2. Now any two elements of H\Ho induce the same automorphism a on Ho. Let x , Y ~ H \ H o be such that x 2 ~ H \ H o . Then, 2a(y),2 -1 = 2 ( x y x - 1 ) 2 -1 = (2x)y(2x)- 1 = ~r(y) = 2y2 - I for all y e H o . It follows that o- = 1 and H (which is generated by H\Ho) is the centralizer of Ho in H. Consequently Ho is abelian and Ho ~ Z(H). We suppose at this point that all elements of H\Ho are involutions. Now x y ~ H \ H o for all x ~ H \ H o , y ~ H o , that is, x y x = y - 1 for all x ~ H \ H o, y ~ H o. But Ho ~ Z(H) and so y = y - 1 for all y s Ho, whence H is elementary abelian of order 8, which is impossible. The last case to consider is when each element of H\Ho is of order 4. Since Ho is abelian and of order 2(2 ~- ~ - 1) it contains a unique involution, But this means that a Sylow 2-subgroup of H has a unique involution and is thus, by Hall [3], p. 189, cyclic or generalized quaternion. Readily we have that H is isomorphic to the quaternion group Q8 of order 8 and Ro is isomorphic to $3. But the semidirect product of these two groups is GL(2, 3). Clearly, ifG is GL(2, 2) then ~ is PG(2, 4). The problem of finding all planes of order 9 of the type we are considering reduces, by Hughes [4], to that of enumerating all 'planar partial difference sets' relative to a complex of five subgroups in GL(2, 3) (one isomorphic to Qs and four to $3). It is sufficient to find all those containing ( ~

01) and (say) e = ( 0- 1

i ) (At this point

the reader is referred to Rahilly and Searby [15] where the problem of enumerating all partial difference sets of Hughes type (6, 2) in PGL(3, 2) is discussed.) Such a partial difference set must have five elements and the author has carried out the necessary search and has obtained eight partial

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difference sets one of which is

01)' (0 11)' ( ; (: : )

10)

Three of the remaining seven are expressible in the form xDy, where x, y e GL(2, 3). Of the remaining four one is D ' = D - l e and the other three are of the form xD'y, x, yeGL(2, 3). The first four yield the nearfield translation plane of order 9 and the other four its dual. Having established Theorem 10, it seems reasonable to seek an analogue for (Qo, Ko)-transitive planes with Qo incident with Ko. The situation in that case is not so simple since many classes of finite generalized Hall planes (that is, finite translation planes with tt Baer subplanes) are known and the class of Ostrom-Rosati planes provides further examples. As an analogue to Theorem 10 we offer the following theorem. T H E O R E M 11. Suppose n is a finite plane of order m2(m >12) that is tt relative to a Baer subplane no and which is (Qo, Ko)-transitive for an incident point~line pair Qo, Ko of no. Let G be the partially transitive group (of Hughes type (5, m)) generated by the (Qo, Ko)-elation group E and the tt group Ro on n o, and Eo the stabilizer of no in E. Then, m = prfor some prime p and positive integer r, E centralizes Eo and Eo and E/Eo are elementary abelian. Furthermore, either E -- Z(E) in which case E is elementary abelian and Ro is isomorphic to the group of linear substitutions on GF(pr) or E o = Z(E) and p is odd. Proof. E is normal in G (Theorem 8 (ii)), G is the semi-direct product of E and Ro and Ro induces (by conjugation) a group of automorphisms on E which fixes Eo pointwise and which is regular on E\E o. Since R o is regular on E\E o and IEl = lEo [2 it follows that [E o I = m = p~, for some prime p and positive integer r. So both E and Eo are p-groups. Now Eo is normal in a larger subgroup of E and so is normal in E itself due to the automorphisms induced by Ro. If m = 2, then rc is PG(2, 4) and so E is C2 x Cz and so the result in this case is trivial. If m > 2, then we can choose x, 2eE\Eo such that x2eE\Eo and show (as in the proof of Theorem 10) that E\Eo centralizes Eo. Since E\Eo generates E we have that E centralizes Eo and so Eo is abelian and, in fact, Eo ~ Z(E). Now yxeE\Eo for all y~Eo, x~E\Eo and so 1 = (yx)pu = yP"xp" = yP~ for all yeEo, where pU is the order of an element in E\Eo. Thus all non-trivial elements of E have the same order, which must be the prime p. In the light of this Eo is elementary abelian. Either Eo = Z(E) or there is xeE\Eo in Z(E). But then E\Eo c_ Z(E) and so E is abelian, whence elementary abelian. If E is elementary abelian, then

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clearly E/Eo is also. So we concentrate on the case Eo = Z(E). Let Z be a subgroup of E such that Eo ~< Z ~< E and Z/Eo = Z(E/Eo). Since E/Eo is a p-group, Z ~ E0 and, since Ro fixes Z, we have that Z = E, that is, E/Eo is abelian, whence elementary abelian. Finally, if E is elementary abelian we can represent it as the additive group of GF(p 2r) and Eo as the additive group of GF(p ~) ~ GF(p2~). Then automorphisms of E fixing Eo pointwise are representable by matrices of the form ( ~

01)over GF(p~). The full group o f t h e s e matrices (of order

m(m - 1)= p,'(pr_ 1)= IRol) is isomorphic to the group of linear substitutions on GF(p"). In the case E0 = Z(E) we cannot have p = 2 since a group consisting entirely of involutions is necessarily abelian.

We are now in a position to decide which Lenz-Barlotti classes contain finite planes possessing a tt Baer subplane. T H E O R E M 12. I f re is a finite plane with a tt Baer subplane rco, then zt is in one of the following Lenz-Barlotti classes: 1.1, II.1, IVa.1, IVa.3, IVb.1, IVb.3, or VIL2. Proof. Jha [7] has shown that any translation plane with a tt Baer subplane is in class IVa.1, IVa.3 or VII.2 and (dually) any dual translation plane with a tt Baer subplane is in class IVb.1, IVb.3, or IV.2. (He also points out that the classes IVa.3, IVb.3 and VII,2 contain only one plane apiecethe planes of Theorem 10.) We shall consider planes that are neither translation planes nor dual translation planes. Such a plane must be in one of the classes 1.1-4, II.1-2. If it is in 1.2-4 or II.2 it is then (Qo, Ko)-transitive for some non-incident point/line pair Qo, Ko. Now Qo, Ko must belong to each tt Baer subplane. But then Theorem 10 shows that the plane is in class IVa.3, IVb.3 or VII.2, which is a contradiction. Thus such a plane is in Lenz-Barlotti class 1.1 or II.1. Theorem 12 is, in one sense, a 'best possible' result because each of the LenzBarlotti classes in the statement is known to contain a finite plane which is tt relative to a Baer subplane (see Jha [7]). It leaves open, however, the problem of the exact composition of classes 1.1, II.1, IVa.1 and IVb.1 insofar as tt planes are concerned. Before proceeding to a brief survey of the known finite tt planes we shall prove a last theorem. T H E O R E M 13. Let 7z be a dual translation plane of order q2 with centre of shears P which is derivable relative to a derivation set E (on a line l~) containing P. Suppose % is a Baer subplane containing E, l~zo~(~c-~\{/~)) such that P is on I. Let ~o be the Baer subplane of the derived semi-translation plane ~ of n whose affine points are those ofl and ~ be the set in ~ which replaces E. Then ff is tt relative to ~o if and only if rt is (P', l, ZCo)-transitivefor all points

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P'6Z\{P}. Further, 77 has a unique tt Baer subplane if and only if l~ is not a translation line of n, except possibly when q = 4. Proof. Suppose n is (P', l, no)-transitive for each P'eE\{P}. The group generated by the (P', l, no)-homologies is of order q2 _ q and fixes E and also fixes I pointwise. The induced group in 77fixes 770 pointwise. Suppose 77 is tt relative to 770 with tt group G(77o). Now G(~o) induces in n a group of central collineations with axis 1 which is of order q2 _ q and which fixes no. It follows that n is (P', l, no)-transitive for all P'~E\{P}. If l~ is a translation line then 77is a translation plane and so is a generalized Hall plane. But then any line ( v~ l~) of n through P yields a tt Baer subplane in rL Suppose there is another tt Baer subplane in 77 distinct from r~o. Let [~ be the line ofr~ corresponding to l~ in n. If there are more than q2 translations in 77with axis [~ then this translation group shifts r~o and fixes Z pointwise. If there are exactly q2 translations with axis To~,then by Theorem 2 of Ostrom [10], we have that 7ooand g are fixed by all collineations of 77,unless possibly q = 4. Either way we have two tt Baer subplanes 77o and 771 containing £. If 77o and 771 share an affine point of 77 then there are collineations in n which shift P and so n is desarguesian. So suppose r7o and 771 share no affine point of 77. These Baer subplanes must share a line rfi of 77 different to l~. But then n is (P', l, rq)- and (P', l', nl)-transitive for all P'eE, where l' is the line of n whose affine points are those of 771 and rq is the Baer subplane of 7c whose affine points are those of rfi. It then follows by applying the dual of a result of Ostrom (see Johnson [8], p. 366) that l~ is a translation axis. Note that the argument of the last paragraph of the proof of Theorem 13 shows that, if n has a translation with axis l~o and centre not P, then lo~ is a translation axis of n. Thus, if l~ is not a translation axis of n, then n is a 'strict' dual translation plane with respect to l~. By Ostrom [10], the full collineation group of 77 fixes l'~ and ,E, unless possibly q = 4. Furthermore, an analogous result holds if l~ is a translation axis of n for, either n is desarguesian, whence ff is a Hall plane and Z is fixed if q > 3, by Hughes [5], or n is a non-desarguesian semifield plane with translation line l~, whence 77is a generalized Hall plane and Y, is fixed, if q > 4, by Rahilly [12]. Notice that the following pertains: (a) if n is desarguesian, then there are qZ(q + 1) tt Baer subplanes containing Z in 77, (b) if n is a non-desarguesian semifield plane with axis lo~, then there are qZ tt Baer subplanes containing Z" in ~, (c) if n is strict with respect to l~, then there is a unique tt Baer subplane containing E in 77. We note in passing that the tt Baer subplanes just enumerated are all the tt Baer subplanes in 77except in certain special cases where q ~<4.

ON T A N G E N T I A L L Y T R A N S I T I V E P R O J E C T I V E P L A N E S

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If zc is a Knuth semifield plane of the type considered by Jha in Part II, Section B of [7] then zr fits the situation under discussion with lo~ not a translation line. The same is true if lr is a derivable proper nearfield dual translation plane. In consequence the 'derived Knuth planes' (or 'DKP's' as Jha calls them) and the derived dual nearfield planes possess a unique tt Baer subplane. For instance, the plane derived from the nearfield dual translation plane of order 9 (that is, the Hughes plane of order 9) is tt relative to a unique Baer subplane, which, of course, is a well-known result. We now see that planes of the type considered in Theorem 7(a) exist. The known partially transitive planes of Hughes type (4, m) are the Hall planes, the Ostrom-Rosati planes and the duals of these. The known planes of Hughes type (5, m) are the generalized Hall planes (for which many constructions are k n o w n - s e e Rahilly [12]), the Ostrom-Rosati planes and the duals of these. There are only two known planes of Hughes type (6, m). They are of order 16, mutually dual and of Hughes type (6, 2). One is a translation plane and the set Z of Theorem 13 is not fixed by the full collineation group. Recently, by an exhaustive computer search for planar partial difference sets in PGL(3, 2), Rahilly and Searby [14] have shown that there are no other planes of Hughes type (6, 2). Some further results concerning the existence of partially transitive planes are contained in [11]. One further recent development is worthy of mention. Peter Lorimer [19] has verified a conjecture of Rahilly and Praeger [11] by showing that, if ~z is a projective plane of Hughes type (6, m) then m = 2 or 3. In Rahilly and Searby [14] this had been shown for m a prime number. In Section 3 of [11] a generalized Hall plane T of order 16 is introduced and in Section 4 the following result (Theorem 1 (i) of [11]) is established: A finite translation plane ~zwith axis l~, of order greater than nine, and not equal to T, is of Hughes type (4, m) if and only if~zis a Hall plane and K0 = l~. Some recent work of N. L. Johnson (see [ 18] ) on the orbits of the collineation group ofT on l~ implies that T is not of Hughes type (4, m) with Qo on loo and K0 4: loo. So Theorem 1 (i) of [11] can be improved to the following complete characterization theorem: A finite translation plane zr with axis lo~, of order greater than nine, is of Hughes type (4, m) if and only if zr is a Hall plane and Ko -- lo~.We remark that, if'(4, m)' is replaced by '(5, m)' and 'Hall' by 'generalized Hall' in the revised statement above, then we obtain the statement of Theorem 1 (ii) of [11]. Finally, we note that Theorem 1 (iii) of [11] characterizes the unique translation plane of Hughes type (6, 2) as the only translation plane of Hughes type (6, m).

ACKNOWLEDGEMENTS

The bulk of the work of this paper was done when the author was visiting the Istituto di Geometria, Universitfi di Bologna, and was supported by

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C N R . T h e a u t h o r w i s h e s t o t h a n k C h e r y l E. P r a e g e r f o r d r a w i n g h i s a t t e n t i o n t o t h e r e s u l t o f E. E. S h u l t u s e d i n S e c t i o n 4.

REFERENCES 1. Baer, R. : 'Homogeneity of Projective Planes'. Amer. J. Math. 64 (1942), 137-152. 2. Baer, R. : 'Projectivities with Fixed Points on Every Line of the Plane'. Bull. Amer. Math. Soc. 52 (1946), 273-286. 3. Hall, M. : The Theory of Groups. Macmillan, New York, 1959. 4. Hughes, D. R. : 'Partial Difference Sets'. Amer. J. Math. 78 (1956), 650-674. 5. Hughes, D. R.: 'Collineation Groups of Non-desarguesian Planes. I. The Hall VeblenWedderburn systems'. Amer. J. Math. 81 (1959), 921-938. 6. Hughes, D. R. and Piper, F. C.: Projective Planes, Graduate Texts in Mathematics 6. Springer-Verlag, Berlin, Heidelberg, New York, 1973. 7. Jha, V. : 'On Tangentially Transitive Translation Planes and Related Systems'. Geom. Dedicata 4 (1975), 457-483. 8. Johnson, N. L. : 'Semi-translation Planes of Class 1-3'. Bol. Un. Mat. Ital. 7 (1973), 359 376. 9. Ostrom, T. G.: 'Transitivities in Projective Planes'. Canad. J. Math. 9 (1957), 389-399. 10. Ostrom, T. G.: 'Collineation Groups of Semi-translation Planes'. Pacific J. Math. 15 (1965), 273-279. 11. Praeger, Cheryl E., and Rahilly, Alan: 'On Partially Transitive Projective Planes of Certain Hughes Types'. Group Theory, Proceedings of a Miniconference, ANU. Lecture Notes in Mathematics 573 Springer-Verlag, Berlin, Heidelberg, New York, pp. 85-111. 12. Rahilly, Alan: 'Finite Generalized Hall Planes and their Collineation Groups'. PhD Thesis, University of Sydney, 1973. 13. Rahilly, Alan: 'Geometry in Generalized Hall Planes'. Geom. Dedicata 5 (1976), 207-217. 14. Rahilly, Alan and Searby, David : 'On Partially Transitive Planes of Hughes Type (6, m)'. Geom. Dedicata, to appear. 15. Rahilly, Alan and Searby, David: 'Partial Difference Sets in PGL (3,2)'. Unpublished. 16. Room, T. G. : 'Geometric Axioms for the Hall Plane'. J. London Math. Soc. (2) 6 (1973), 351-357. 17. Shult, E. E. : 'On a Class of Doubly Transitive Groups'. Illinois J. Math. 16 (1972), 434-445. 18. Johnson, N. L. : 'A Note on the Derived Semifield Plane of Order 16'. Aequationes Mathematicae 18 (1978), 103 111. 19. Lorimer, Peter: 'On Projective Planes of Type (6, m)'. Report Series No. 156, University of Auckland, November, 1979.

Author's address:

Alan Rahilly, School of Applied Science, Gippsland Institute of Advanced Education, Churchill, Hetoria 3842, Australia (Received February 28, 1980)