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Consider a finite field $F_p$ with $p$ elements; how would one calculate the number of lines and the number of planes in the affine space $F^3_p$?

If one knew the number of lines through a particular point, the number of planes could then be calculated by multiplying the number of lines through the origin by the number of points on any line.

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1 Answer 1

There are $p$ points on every line. The lines through a given point can go through every other point, and two distinct ones have only that point in common, so there are $$ \frac{p^{3} - 1}{p-1} = 1 + p + p^{2} $$ lines through a point.

If you count the number of lines as $$ \text{number of points} \cdot \text{number of lines through each point} = p^{3} \cdot (1 + p + p^{2}), $$ then you are counting each line $p$ times, one for each of its points, so the number of lines is $$ p^{2} \cdot (1 + p + p^{2}). $$


Gerry Myerson (thanks!) made me notice that I had forgotten to count planes.

One way is the following. Count first the triples of distinct, non-collinear points. Their number is $$ p^{3} (p^{3} -1) (p^{3} - p). $$ To count planes, we have to divide by the number of triples of distinct, non collinear points on a given plane, that is $$ p^{2} (p^{2} -1) (p^{2} - p). $$ The net result is $$ \frac{p^{3} (p^{3} -1) (p^{3} - p)}{p^{2} (p^{2} -1) (p^{2} - p)} = p (p^{2} + p + 1). $$ The same method allows for an easier counting of the lines, as $$ \frac{p^{3} (p^{3} - 1)}{p (p-1)} = p^{2} (p^{2} + p + 1). $$

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And the number of planes? –  Gerry Myerson Jun 11 at 9:28
    
@GerryMyerson, you're right, I missed that one. –  Andreas Caranti Jun 11 at 10:34
    
@GerryMyerson, added, thanks a lot. –  Andreas Caranti Jun 11 at 16:02

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