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Peter Lorimer, an Introduction to Projective Planes

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Peter Lorimer, an Introduction to Projective Planes AN INTRODUCTION TO PROJECTIVE PLANES : SOME OF THE PROPERTIES OF A PARTICULAR PLANE OF ORDER 16 Peter Lorimer Dedicated to H.G. Forder on his 90th birthday (received 25 July, 1979) As my first introduction to projective geometry was in Professor Forder's third year class at the University of Auckland it is a great pleasure to be able to contribute a paper on projective planes to this volume on his 90th birthday. My intention here is to discuss a certain projective plane of order 16 in a way that should be accessible to the uninitiated. It is not one of the classical Desarguesian planes that we learnt about from Professor Forder : its interest is less geometrical, more algebraic and combinatorial. Within the space of a few years the existence of a particular projective plane of order 16 was discovered independently by three people. At the University of Iowa, N.L. Johnson observed [3] that a certain semifield plane is derivable; in his thesis at the University of Sydney, A. Rahilly described a generalized Hall plane; and in 1972, while I was on sabbatical leave from the University of Auckland, I discovered an explicit construction for a plane. That the three planes are identical is not obvious : it is clear in Rahilly's paper [7] that his plane is derivable from a semifield plane but I have the embarrass­ ment of having stated in my paper [6] that mine is not. The identity problem was cleared up by Johnson and T.G. Ostrom [4] who proved that Math. Chronicle 9(1980) 53-66. 53 both these planes are tangentially transitive with respect to a sub­ plane of order 2, and that there is only one translation plane of order 16 with this property. Translation planes which are tangentially transitive were being studied at Westfield College of the University of London, mainly by V. Jha. He characterized all tangentially transitive translation planes of order not 16 [2] and M. Walker completed Jha's work on planes of order 16 by proving that there is only one which is tangentially transitive with respect to a subplane of order 2, thus obtaining independently the result of Johnson and Ostrom mentioned in the last paragraph. Jha's work, supplemented by these other two papers, shows that there is only one finite translation plane which is tangentially transitive with respect to a subplane of order different from the square root of its own order. Walker [8] also exhibited another translation plane of order 16 with properties similar to the one which is the subject of this paper: for example the simple group of order 168 acts on both as a collineation group. Johnson and Ostrom [5] proved that these are the only two translation planes of order 16 on which this group acts. Another feature of the present translation plane is that it is of type (6,2) in the terminology of D. Hughes [l]. This is pointed out by Johnson and Ostrom [4] who remark that it appears to be the only known projective plane of class (6,m) , for any m . The construction of a plane of order 16 The projective plane of order 2 has 7 points and 7 lines: if the points are 1,2,3,4,5,6,7 the lines can be taken as the subsets {1,2,3), {3,4,5}, {5,6,1}, {1,4,7}, {3,6,7}, {2,4,6}. For our purposes the main properties of this structure are 54 PI. Each two points are joined by a unique line. P2. Each two lines meet in a unique point. P3. Among the points there are four, no three of which lie together on a line. A structure of points and lines satisfying these three axioms is called a projective plane; to emphasize the fact that there do exist structures other than the classical projective planes, the adjective non-Desarguesian is used for planes in which Desargues' Theorem is not true. Associated with each line {a,b,o} of the above plane of order 2 are two 3-cycles, {aba) and {aob): these are to be regarded as permuta­ tions of the three points involved and they do not act at all on the other 4 points of the plane. The symbols 0,1 are added to the 14 3-cycles which come from the 7 lines of the plane to form a set M = {0,1, (123), (132) , (345) , (354) , (561) , (516) , (147) , (174) , (257) , (275) , (367),(376),(246),(264)) The object now is define addition and multiplication on M in a way that is close enough to the way that these operations are defined on the real numbers that the usual techniques of analytic geometry can be used to construct the geometric structure we are after. First, addition and multiplication are defined on each set {0, 1, {aba), {aob)} in such a way to make it a field of order 4. Second, suppose that {pqr) and (pat) are 3-cycles coming from different lines of the plane of order 2. (because the two lines are from a projective plane they have exactly one point, p , in common). The definitions are {pqr) + {pet) = (puv) where u is the third point on the line joining q and 8 ; 55 (pqr)(pst) = (tru) where u is the third point on the line joining t and r . For example (123) + (147) = (165) (123)(147) = (736) In all this notice that a 3-cycle (xyz) can equally be written as (yzx) or (zxy): hence also (246) + (354) = (471) after orienting the left hand side to (462) + (435). The basic algebraic properties of M with these two binary operations are given by Theorem 1. (1) (M,+)is a 4-dimensional vector space over the field of order 2 (or alternatively it is an elementary abelian group of order 16) (2) if xsysz € M then x(y + z) = xy + xz (3) if x,y £ M and xz = yz for some z t 0 then x = y. The proof of this theorem is a straightforward verification of the three statements. Although the properties mentioned in the theorem closely mimic those of a field, M is not itself a field: for example the laws (a: + y)z = xz + yz xy = yx are not always true. Now let us set about constructing a plane from M . The real plane of co-ordinate or analytic geometry is constructed from the real 56 numbers JR by taking the members of J? x J? as points and the lines as solutions of linear equations. This is the method that is used here, too. Among the points of the projective plane of order 16 are to be 256 points which are identified with the members of the set M x M written as ordered pairs in the form (x,y) , in the usual manner. If m,a are both members of M there is to be a line consisting of all Cx,y) ( II x (I for which y = mx + a . This line will be denoted by [w,e], the square brackets distinguishing lines from points: thus the point (x,y) lies on the line [/n,c] if and only if y = mx + o . In addition, for each member d of M there is to be a line, denoted by [<2] consisting of the 16 points (d,y) with y in M . At this stage we have carried out the construction suggested by the way that the usual real plane is constructed from J? . But it is not a projective plane; rather it is an affine plane, characterized by the axioms: A1 : Each two points are joined by a unique line. A2 : Given a line and a point not on the line, there is exactly one line which passes through the given point, but does not meet the given line. A3 : Among the points there are four, no three of which lie together on a line. There is a standard way of constructing a projective plane from an affine one. Here is how it works in the present case. To the lines 57 already constructed, add one more; call it the line at infinity and denote it by ["]. To the points already constructed add seventeen more; denote one of them by («) and the other 16 by the sixteen symbols (a) where z is a member of M . Each of these 17 points is to lie on the line at 00 ; in addition the point (m) is to lie on each of the 16 lines [m,c], c ( M; and («*>) lies on each of the lines [<3], d * M . If we were to use the language of ordinary analytic geometry we would say that the point (ro) lies on each line of slope m , m being the slope of the line [ot,c?]; hence it makes sense to think of (m) being at infinity, or on the line at infinity. This completes the construction of the projective plane: it has order 16 because there are 17 points on every line (in the affine plane first constructed there were 16 points on every line). From now on this plane will be denoted by n . The collineatlon group of the plane A collineation of a projective plane is a 1-1 mapping of the points onto the points which maps the set of points on any line onto the set of points of another (or possibly the same) line; alternatively it is a 1-1 mapping of the lines onto the lines which maps the set of lines through any point onto the set of lines through another (or poss­ ibly the same) point.
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