Galois Planes

A class of projective planes called Galois planes is constructed as follows. Let 86#35 be a finite field of order 87#36. Let 88#37 be the 3-dimensional space over 86#35. We define the inner product over 88#37 in the usual way: for 89#38 and 90#39 we set 91#40. We say that 92#41 and 93#42 are perpendicular if 94#43.

Let us say that two nonzero vectors 95#44 are equivalent if 96#45 for some 97#46.

Let 98#47 be the set of equivalence classes on 99#48. Note that each equivalence class has 100#49 elements and therefore the number of equivalence classes is 101#50.

Set 102#51 and let us say that 74#23 and 76#25 are incident if 94#43 where 103#52 (92#41 is a vector in the equivalence class 62#11) and 104#53. The coordinates of 92#41 are called homogeneous coordinates of 62#11 (they are not unique-every point has 100#49 triples of homogeneous coordinates); similarly, the coordinates of 93#42 are called homogeneous coordinates of $\ell$.

105#54

106#55

A set of points is collinear if there is a line with which all of them are incident.

We say that four points are in general position if no three of them are collinear.

A collineation is a transformation of the projective plane consisting of a permutation of the points and a permutation of the lines which preserves incidence.

Theorem 5.14 (Fundamental Theorem of Projective Geometry.)   If 107#56 and 108#57 are two quadruples of points in general position in a Galois plane then there exists a collineation 109#58 such that 110#59.

111#60

112#61

113#62

114#63

115#64

116#65

Puzzle 5.15   Find two infinite sets of nonnegative integers, 117#66, such that every non-negative integer can be written in a unique way as 118#67, with 119#68 and 120#69.

Note:

(a)

We necessarily have that 121#70.

(b)

If $A$ is allowed to be finite, then we could take, for our favorite number 122#71, 123#72 and 124#73. The hard part is to make both $A$ and 125#74 infinite.