Subgroups of a finite elementary abelian group.

Question

I am looking for a method to calculate number all subgroups of a finite elementary abelian $p$-group.

Suppose $G$ be an elementary abelian $p$-group of order $p^n$. A proper subgroup $H$ of $G$ is also an elementary abelian $p$-group of order $p^r$ where $r<n$.

We can realize $G$ as $n$ dimensional vector over $\mathbb Z_p$ and number of subgroups of $G$ of order $p^r$ is equal to the number of $r$ dimensional subspaces of the vector space.

Now we have to count the number of linearly independent $r$-tuples of vectors but several tuples may give same subspace.

Now what should I do??

Answer 1

I am thinking about the Transitive (?) group action of $GL_n(\mathbb F_p)$ on the set of linearly independent $r$-tuples. Number of orbits of this action will give the number of $r$-dimensional subspaces and hence the number of subgroups of order $p^r$. Is my this approach correct??