Elementary Abelian $p$-Subgroups of $GL_n(\mathbb{F}_p)$ Let $p$ be a prime number.  If $G$ is a finite group, an elementary abelian $p$-subgroup of $G$ of rank $r$ is any group $E\subset G$ such that $E\cong (\mathbb{Z}/p)^{\oplus r}.$
Let $\mathcal{E}_r$ be the collection of all elementary abelian $p$-subgroups of rank $r$ of the finite group $G=GL_n(\mathbb{F}_p).$  Is anything known about $\mathcal{E}_r$?  I would be happy with a reference, or an answer to any of the following related questions:


*

*What is $|\mathcal{E}_r|$?

*How is $\mathcal{E}_r$ partitioned into orbits under the conjugation action of $G$?

*How many elements of $\mathcal{E}_r$ lie in a fixed Sylow $p$-subgroup of $G$?


Every element of $GL_n(F)$ of order a power of $p=\operatorname{char} F$ is unipotent, meaning its characterisitc polynomial is $(X-1)^n$. It is therefore conjugate in $GL_n(F)$ to an upper triangular matrix with entries $1$ on the main diagonal, and conjugacy classes of such elements are characterised by their common Jordan type, a partition of $n$. The actual order of an element is the least power of $p$ that is no less than the size of the largest Jordan block, so to have order $p$ all Jordan blocks should be of size at most$~p$. (In particular if $n\leq p$, then the set of all unitpotent upper triangular matrices, which is always a Sylow $p$ subgroup, contains non-identity elements of order $p$ only.)
This was for individual elements, but for two $p$-elements to commute, they must be simultaneously trigonalisable; therefore every elementary $p$-subgroup is conjugate to an elementary $p$-subgroup of the upper triangular matrices (alternatively use that every $p$-subgroup is contained in a Sylow $p$-subgroup). This still answers none of your questions, but should help getting under way.
