Let $p$ be a prime number and $G=GL_n ( \mathbb{Z} / p \mathbb{Z} )$. Consider the set $U$ of upper-triangular matrices of $G$ having entries of $1$ on the diagonal. The cardinality of $U$ is $p^{\frac {n(n-1)} 2}$ and $U$ is a subgroup of $G$, in particular $U$ is a Sylow $p$-subgroup of $G$. It is well known that the Sylow $p$-subgroups of a group $G$ are conjugate, and every $p$-subgroup $H$ of $G$ is contained in some Sylow $p$-subgroup of $G$. Then there exists $g\in G$ such that $H\leq gUg^{-1}$, which allows us to compute the number of conjugacy classes of elementary abelian subgroups of rank $2$ ($H=( \mathbb{Z} / p \mathbb{Z} ) ^2$) in the Sylow $p$-subgroup $U$. Any help would be appreciated so much. Thank you all.


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