Equivalence of definitions of size of projective space

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Statement

For a vector space over a prime field, or an elementary abelian group

Suppose p is a prime number and V is an elementary abelian group of order p^k, i.e., a vector space of dimension k over the field of p elements. Then, the following three sets have the same size:

  1. The set of minimal subgroups of V, i.e., subgroups of order p.
  2. The set of maximal subgroups of V, i.e., subgroups of order p^{k-1}.
  3. The projective space for V (this is a projective space of dimension k - 1 over the field of p elements).

Moreover, the size of all three sets is:

\frac{p^k - 1}{p - 1} = p^{k-1} + p^{k-2} + \dots + 1

For a vector space over a finite field

Suppose q is a prime power and V is a vector space of dimension k over the field with q elements. Then, the following three sets have the same size:

  1. The set of one-dimensional subspaces of V.
  2. The set of codimension one subspaces of V.
  3. The projective space for V (this is a projective space of dimension k - 1 over the field of q elements).

Moreover, the size of all three sets is:

\frac{q^k - 1}{q - 1} = q^{k-1} + q^{k-2} + \dots + 1