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 set $U$ is a Sylow $p$-subgroup of $G$. I search to compute the number of commuting pairs in the p-group $U$. It is well known that this number is $|U|·k(U)$ , where $k(U)$ is the number of conjugacy classes of $U$. Then the question is equivalent to determining the number of conjugacy classes of $U$. Which of the two proposals is correct? and why:

• $k(U)$ is the index of its normalizer in $G$.

• $k(U)$ is given by Higman Conjecture.

Any help would be appreciated so much. Thank you all.