Galois geometry

Number of subspaces of a Galois geometry

The Galois geometry contains subspaces of dimension , fin where

is the Gaußian binomial coëfficient.¹

Proof

We define

Let , so . As a vector space contains points. Now the number of points in equals the number of 1-dimensional subspaces of , and different subspaces have only the origin in common. Thus the number of 1-dimensional subspaces is

Each -dimensional subspace is determined by linearly independent points, so the number of -dimensional subspaces in an -dimensional projective space is the total number of independent sets of points of size divided by the number of independent sets of points of size in each -dimensional subspace. This gives

as required.

---

tidy | en | sembr

Footnotes

1. 2020. Finite geometries, ¶4.7, p. 79 ↩