311 – Extra credit exercise

This optional exercise is due Thursday, February 12, at the beginning of lecture.

Show that the projective plane \mathbb Z_2P^3 obtained through the vector space construction applied to the field \mathbb Z_2 of two elements is (isomorphic to) the Fano plane.

I review the construction: To form KP^2 we start with a field K and consider K^3.

A point in the projective plane is just a line in K^3 through the origin. Note that any such line is obtained by fixing a non-zero element (x,y,z) of K^3 and considering all its scalar multiples, all the vectors (\alpha x,\alpha y,\alpha z) with \alpha\in K.

(When K=\mathbb Z_2, the only values of \alpha are 0 and 1, so a line consists of only two points, one of which is the origin.)

A line in the projective plane is actually the collection of lines contained in a plane in K^3 through the origin. In other words, a set L of points (that is, of lines through the origin) is a line iff there is a plane in K^3 that contains the lines that are the elements of L and no others. Note that any such plane can be described by fixing coefficients \alpha,\beta,\gamma in K, not all equal to zero, and considering the set of vectors (x,y,z)\in K^3 such that \alpha x+\beta y+\gamma z=0.

To show that when K=\mathbb Z_2 this construction produces the Fano plane it is not enough to indicate that the resulting space consists of 7 points and 7 lines, you also need to verify that these can be identified with the points and lines of the Fano plane in such a way that all incidence relations in the Fano plane are verified in this space and no, others.

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