## 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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