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A Translation Plane of Order 81 and Its Full Collineation Group

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A Translation Plane of Order 81 and Its Full Collineation Group BULL. AUSTRAL. MATH. SOC. 50D30 VOL. 29 (1984), 19-34. A TRANSLATION PLANE OF ORDER 81 AND ITS FULL COLLINEATION GROUP VITO ABATANGELO In this paper a new translation plane of order 8l is constructed. Its collineation group is solvable and acts on the line at infinity as a permutation group K which is the product of a group of order 5 belonging to the center of K with a group of order U8 . A 2-Sylow subgroup of K is the direct product of a dihedral group of order 8 with a group of order 2 . K admits six orbits. They have lengths k, 6, 12, 12, 2k, 1. Introduction Let GF(q) be a Galois field of odd order q (> 5) and let F be the regular spread of PG(3, q) • Suppose that F contains a set R of reguli satisfying the properties R (i) R consists of (<?+3)/2 reguli, R±, i?2> ..., (q+^y2 > (ii) any two reguli of R have exactly two lines in common, (iii) no three reguli of R have a line in common. Received 2 September 1983- Research partially supported by MPI (Research project "Structture Geometriche Combinatorie e Loro Applicazioni")• The author wishes to thank G. Korchmaros for the discussion with him about the argument of the present paper. Copyright Clearance Centre, Inc. Serial-fee code: 0OOU-9727/8U $A2.00 + 0.00. 19 Downloaded from https://www.cambridge.org/core. IP address: 170.106.33.22, on 01 Oct 2021 at 16:07:53, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700021237 20 Vito Abatangelo Let U be the partial spread consisting of all the lines of the reguli R^ [i = 1, 2, ..., {q+3)/2) and let JR denote the point set covered by U . If S! denotes the opposite regulus of R. , the set U' of all the lines of R'. is not a spread of I- because any point P of I~ belongs to two lines of U' . According to a result of Bruen [4], if a subset V of U' is a partial spread then V contains (q+l)/2 lines of each R'- such that the union of these half-reguli is V . If there exists such a partial spread V then one can obtain a new spread G = (F-U) <•> V by replacing U with V . G will be called 3-derived from F (with respect to {U, V) ) . We can try to obtain a new partial spread V = hV of Xg by choosing only a suitable half of the lines of each opposite regulus. As for 6-derived spreads the existence problem is yet unsolved except for q = 5, 7, 11, 13 • Specific examples are given in Bruen's paper [4] when q = 5, 7 , in [5] and [72] when q = 11 and in [73] when q = 13 - In [4] Bruen also proved that if q - 7 the collineation group of a B-derived spread is the inherited group, that is, it is the subgroup of the collineation group of PG(35 q) which leaves F invariant. The collineation groups of the translation planes arising from the above mentioned examples are determined [5], [72], [73]. In this paper the first example of a 3-derived spread is constructed for q = 9 and the collineation group of the corresponding translation plane is determined. It is a solvable group satisfying the properties listed in the summary. 2. Notation and terminology Notation and terminology are the same as in Bruen's paper [4]. The reader is assumed to be familiar with the theory of projective planes given in Dembowski [6] and in Hughes and Piper [7] and with the theory of permutation groups given in Wielandt [74]. Let GF(9) be the quadratic extension of GF(3) = {0, 1, 2} . In particular let us consider the polynomial x + x + 2 (which is irreducible in GF(3) 1, i { GF(3) such that i + i + 2 = 0 and Downloaded from https://www.cambridge.org/core. IP address: 170.106.33.22, on 01 Oct 2021 at 16:07:53, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700021237 Translation plane of order 81 21 3 = £ , U = i + 1 , 5 = i + 2 , 6 = 2£ , 7 = 1 + 2i , 8=2+ 2i ; so GF(9) = {0, l, 2, 3, U, 5, 6, 7, 8} . The homogeneous coordinates (x, y, z, u) denote points in PG(3, q) • The line joining the point (x , y , z , u ) to the point (^2, y2, *2, «2) is denoted by { (a^, j^, s;L, u±) , (a^, i/2, s2> «2) ) ; moreover if g, ht ..., w are elements of a group, then the subgroup generated by g, h, , w is denoted by (g, h, ...,U> . 3. Preliminaries Before the example, the general relation between the spread F and the Miquelian inversive plane M{q) over GF(<?) is discussed briefly. Assume that q is odd and denote by s a non-square element of G?{q) , so GF[q2) = {a+bt \ a, b € GF(<7), t2 = s} . We define norm N(a+bt) of a + bt of GFfa2) , N(a+bt) = a2 - sb2 . Clearly N(a+bt) € GF(c?) ; moreover set ff(p) = {a+bt \ N(a+bt) = p} . M{q) is considered the incidence structure whose points are the elements of GF[q ) u {°°} and whose circles are the subsets of GF(<7 ) u {°°} of the following types: (I) a + bt + ff(p) = {x+yt | N{(a-x)+(b-y)t) = p} ; (II) C(a+bt, e+dt) = {{a+bt)+\(e+dt) \ X € GF(q) o {<»}} . The spread F is the union of all the lines <(1, 0, 0, 0), (0, 1, 0, 0)>, <(a, sb, 0, l), (b, a, 1, 0)> where a, b run over GF(<7) . By a theorem of Bruck's (C3], also [4]), the map ~ -<(1, 0, 0, 0), (0, 1, 0, 0)> a + bt •* < (a, sb, 0, l), (fc, a, 1, 0) > is an incidence preserving isomorphism between M(q) , with its points and circles, and F , with its lines and reguli. By Bruck's theorem, the problem of finding sets R of reguli satisfying the properties (i), (ii), (iii) is equivalent to that of constructing chains C of circles in M(q) satisfying the properties Downloaded from https://www.cambridge.org/core. IP address: 170.106.33.22, on 01 Oct 2021 at 16:07:53, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700021237 22 Vito Abatangelo 1 (i) C consists of (<7+3)/2 circles, C±, C^ ..., (ii)1 any two circles of C have exactly two points in common, (iii)' no three circles of C have a point in common. Let W be the collineation group of PG(3, q) mapping F onto itself. Every W € W determines a map w' of M{q) onto itself defined as follows: let P be any point of M(q) and let r be the line of F which corresponds to P by V ; then W(P) is the point which corresponds to w(r) by 4* . Therefore a map $ is defined putting $(w) = w' . Of course $ is a homomorphism of W into W , where W denotes the automorphism group of M(q) . The kernel of $ is the group M of all collineations of PG(3, <j) fixing F linewise. M is a cyclic group of order q + 1 . Moreover W o* TL(2, c?n and V ^ PrL(2, q^ . Let fl be the subgroup of W which leaves invariant a set R of reguli satisfying the properties (i), (ii), (iii). If H' denotes the subgroup of W which leaves invariant the chain C of circles of M(q) corresponding to R by ¥ , then *(ff) = H' . Hence ff/Af cs fl' . 4. Two partial spreads U and K of PG(3, 9) covering the same point set Using the notation of Bruen's paper, our construction can be described as follows. If s = 3 , the norm-classes of GF(9) are N{0) = {0} , N(l) = {1, 2, k+kt, 8+Ut, 3+5*, 6+5*, 3+7*, 6+7*, ^+8*, 8+8*} , #(2) = {5, 7, U+t, 8+t, U+2*, 8+2*, 3+3*, 6+3*, 3+6*, 6+6*} , = (5+3*, 7+3*, 3+U*, 6+1**, 5*, 5+6*, 7+6*, 7*, 3+8*, 6+8*} , = {3+*, 6+*, 3+2*, 6+2*, 3*, 1+5*, 2+5*, 6*, 1+7*. 2+7*} , »(5) = ft, 8, 1+3*, 2+3*, 5+5*, 7+5*, 1+6*, 2+6*, 5+7*, 7+7*} , = {*, 2fc, U+3*, 8+3*, l+l»t, 2+ltt, U+6*, 8+6*, 1+8*, 2+8*} , = {3, 6, 1+t, 2+*, 1+2*, 2+2*, 5+1**, 7+1**, 5+8*, 7+8*} , N(Q) = (5+*, 7+*, 5+2*, 7+2*, kt, 1++5*, 8+5*, U+7*, 8+7*, 8*} . An easy calculation shows that the following circles of M(q) form a chain C satisfying the properties (i)', (ii)', (iii)1: Downloaded from https://www.cambridge.org/core. IP address: 170.106.33.22, on 01 Oct 2021 at 16:07:53, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700021237 Translation plane of order 81 23 Cx : 2* + i»(7) , C2 : t + ff(7) , ^ : 7* + »(5) , Cu': 5* + »(5) , C5 : (0, 3+*) , C^ : (0, 3+2*) .
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