Equivalence of definitions of size of projective space

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:

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