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Collineation Groups of Generalized André Planes

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Collineation Groups of Generalized André Planes COLLINEATION GROUPS OF GENERALIZED ANDRÉ PLANES DAVID A. FOULSER 1. Introduction. In a previous paper (5), I constructed a class of translation planes, called generalized André planes or X-planes, and discussed the associated autotopism collineation groups. The main question unanswered in (5) is whether or not there exists a collineation rj of a X-plane II which moves the two axes of II but does not interchange them. The answer to this question is "no", except if II is a Hall plane (or possibly if the order n of II is 34) (Corollary 2.8). This result makes it possible to determine the isomorphisms between X-planes. More specifically, let II and II' be two X-planes of order n coordinatized by X-systems Q and Q', respectively. Then, except possibly if n = 34, II and II' are isomorphic if and only if Q and Q' are isotopic or anti-isotopic (Corollary 2.13). In particular, II is an André plane if and only if Q is an André system (Corollary 2.14). The X-planes II of (5) are very similar to the planes constructed by Ostrom (8; 10; 11 ; 12) by "replacing" certain nets in a Desarguesian plane II* (see § 3). The results in this paper concerning X-planes are derived as a result of inter­ preting X-planes from Ostrom's point of view. In particular, from a lemma of Ostrom (Theorem 2.2), it follows that any collineation of a X-plane II has the form rj = âi-T, where J is a collineation of the underlying Desarguesian plane II*, and â and f are mappings determined by field automorphisms (Theorem 2.5). The results listed above, as well as the results of (5) concerning the collineation group of II, follow directly from this form for 77. There is one case in = 34) to which the proofs below do not apply. It seems probable that this case is not an exception to the results of this paper (see the remarks following the proof of Theorem 2.7). I would like to express my appreciation to T. G. Ostrom for several helpful conversations concerning his unpublished work at a recent conference in Chicago. 2. The collineation group of II (Qx). In order to define the class of generalized André systems, or X-systems (5), we introduce the following notation. Let q = ps, for p a prime, let n = qd, and let/ = ds. Let F = GF(n) and let K = GF(g). Let co be a primitive root of F, let a be the automorphism of F of order/ defined by a: x —> xp, and let p = as so that p has order d. Let Received November 6,1967 and in revised form, March 26,1968. This research was partially supported by NSF Grant GP-6539. 358 Downloaded from https://www.cambridge.org/core. 28 Sep 2021 at 15:59:46, subject to the Cambridge Core terms of use. ANDRÉ PLANES 359 V be the vector space of ordered pairs (x, y) of elements of F. Thus, V is a two-dimensional vector space over F, a 2d-dimensional space over K, and a 2/-dimensional space over GF(p). If a is an automorphism of F/K (i.e., if the fixed field of a contains K), let â be the following mapping of V, â: (x,y) -> (x, a(y)). Let X be an arbitrary mapping from F* (the set of non-zero elements of F) into (p), the automorphism group of F/K. If X(œk) = p\ it is sometimes convenient to write X(k) = t, i.e., to represent X as a mapping from In-\ into Id, where Iz = {0, 1, . , z — 1} is the set of least positive residues (mod z). It is also convenient to define the index of an element m G F* as follows: if k T m — œ w ith k G In-u then ind(m) = k. For a, b £ F, define the binary operation "o" by a o b = apX(a)(6), if a ^ 0, and 0 o b = 0. Finally, let Q be the system Q = {F, +, o}. LEMMA 2.1. Q is a (left) quasi-field (or Veblen-Wedderburn system) if and only if the mapping X: JTw_i —> Id satisfies the following conditions: (a) X(0) = 0; (b) // i, j G In-u and if i =j (modq* — 1), where t = (d, \(i) — X(j)), then i = j. Proof. See (5, Lemma 2.1). If X satisfies the conditions of the lemma, then Q = Q\ will be called a generalized André system, or a \-system. It is always possible to assume (after normalizing X) that the X-system Q\ has Kern K = GF(q) (5, 2.5). For the X-system Qx, define v to be the least integer in /w_i such that X is periodic (mod v); and define u — LCM(g* — 1), where t\d and / < d. Then v\u from Lemma 2.1(b). A X-system Q\ with Kern K is an André system if and only if i = j (mod q — 1) implies \(i) = \(j), for i, j Ç 7w-i; i.e., if and only if v\q — 1 (5; § 3, example 1). The vector space V can be made into the affine plane 11(C) coordinatized by a X-system Q, by defining the points of II (Q) to be the elements (x, y) of V, and the lines of II (Q) to be the solution sets of the equations y = m ox + b and x = c, for m, b, c £ Q\ these lines will be denoted by [m,b] and [c], respectively. The class Ë of lines of II(Q) through the origin, 0 = (0, 0), consists of the lines [m, 0] and [0], for m G F. II(<2) can be completed to a projective plane by adding the points (m) and 7 (oo),mÇ T , °° $ JP, and the line lœ = [<» ], in the usual manner. Let II* be the Desarguesian affine plane defined by F as a two-dimensional vector space over F. That is, the points of II* are the elements of V; and the lines of II* are the solutions of the equations y = mx + b and x = c (for m, b, c G F), which will be denoted by [m, b]* and [c]*, respectively. The class S* of lines of II* through (0, 0) consists of the lines [w, 0]* and [0]*, m G i7. In the projective completion of II*, the points (m)* and (°°)*, and the line Zœ* = [°° ]* are added to II*. Note. The "*" will distinguish the Desarguesian plane II* and its lines, from the X-plane II (Q) and its lines. Downloaded from https://www.cambridge.org/core. 28 Sep 2021 at 15:59:46, subject to the Cambridge Core terms of use. 360 DAVID A. FOULSER THEOREM 2.2 (12, Lemma 1). Let rj be a non-singular semi-linear trans­ formation of V over K. Let SDt, be the set of those lines of S* which are mapped onto lines of Ê* under rj. Finally, let t be the maximal proper divisor of d. If 12^1 > Q* + 1> in particular if |5D?J > qd/2 + 1, then 77 is a Desarguesian collineation of II* {i.e., 77 is semi-linear over F). Proof. The proof follows from Lemmas 2.3 and 2.4. LEMMA 2.3. 7f |9)î,| ^ 3, then |3W,| = g' + 1 /0f ^wg / which divides d. LEMMA 2.4. Assume that 77 (w (2.2)) fixes the Desarguesian lines [0, 0]*, [1, 0]*, and [0]*. Let Mv = [m G i?: [w, 0] G 2W,}. Then Mv is a subfield of F which contains K. Proof. Since 77 fixes the subspaces y = 0, x = 0, and j = x of F, then 77 as a 2d-dimensional semi-linear transformation over K has the form 77: (x,y) -» (o-(x), c(y)), where o- is a ^-dimensional semi-linear transformation of F over K. The mapping of F, in: x —» wx, is a linear transformation of F, for m G F. For m G M„ let k be defined by 77: [m, 0]* —» [k, 0]*. Then 77 (x, wx) = rj (x, m{x)) = (<r(x), <rin{x)) = (o-(x), fe<r(#)); thus k = afh(j~l. If m, G M„ with 77: [ra<, 0]* —» [&*, 0]* for i = 1, 2, then Wi + m2 G -M", and mim2 G -M",. For example, to show that WiW2 G Af„ consider the line \m\m<i, 0]*. Applying 77, 77(x, m\m2x) = (o-(x), aMim2{x)) = (o-(x), m^mi^x)). Since m£ = kh i = 1, 2, then mfm<£ = k1k2 = &1&2. Hence, 77: [mun2, 0]* —> [kik2, 0]*; thus Wiw2 G Mv. Therefore, Mv is a subfield of F. Since cr is a semi-linear trans­ formation over K, K Cl Mv. To prove Lemma 2.3, assume that |3W,| è 3. Let lt and // for i = 1, 2, 3 be three lines of SO?,, and their images under 77, respectively. Since these six lines are Desarguesian lines, there exist Desarguesian collineations n and r2 of II* such that n: [0,0]*^/x, [0]*->/2, [l,0]*->/3 and r2: W -> [0, 0]*, U -> [0]*, W -» [1, 0]*. Further, let 77' = T277n. Thus, 77' satisfies the hypothesis of Lemma 2.4. More­ over, 77' maps a Desarguesian line / onto some other Desarguesian line if and only if 77 maps the Desarguesian line n(/) onto a Desarguesian line; i.e., |2»,| = |2K,'|. By Lemma 2.4, |2K,'| = \Mj\ + 1 = g' + 1, where t\d.
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