Let $p$ be a prime number, and let $k=\mathbb{F}_p$ be the field of $p$ elements. Let $G=GL_n(k)$. We know that
$$|G|=\prod_{i=0}^{n-1}(p^n-p^i)=p^{\binom{n}{2}}\prod_{i=0}^{n-1}(p^{n-i}-1)$$
so that the Sylow $p$-subgroups of $G$ have order $p^{\binom{n}{2}}$. One such subgroup is $U$, the upper-triangular unipotent subgroup consisting of all upper-triangular matrices with $1$'s on the diagonal. Let $A_{ij}=I_n+E_{ij}$ for $j>i$, where $E_{ij}$ is the matrix with a $1$ in the $ij$th entry and $0$'s elsewhere. If we find $i_1,\ldots,i_r,j_1,\ldots,j_r$ such that the $A_{i_kj_k}$ pairwise commute, then we have:
$$(\mathbb{Z}/p\mathbb{Z})^{\oplus r}\cong \langle A_{i_1j_1},\ldots,A_{i_rj_r}\rangle\subset U$$
Two questions:
- Is every copy of $(\mathbb{Z}/p\mathbb{Z})^{\oplus r}$ inside of $U$ conjugate under $G$ to a subgroup of the form $\langle A_{i_1j_1},\ldots,A_{i_rj_r}\rangle$?
- Do there exist distinct, conjugate subgroups of the form $\langle A_{i_1j_1},\ldots,A_{i_rj_r}\rangle$ and $\langle A_{i_1'j_1'},\ldots,A_{i_r'j_r'}\rangle$?
The largest value of $r$ for which such subgroups exist is $r=\lfloor n^2/4\rfloor$. In this case, I believe the answers to these questions are yes and no, respectively. I'd like to know if this also holds for smaller values of $r$.