Let $U_1$ be the group of $n\times n$ upper triangular matrices with 1's down the main diagonal (called unit upper triangular matrices) over $\mathbb{F}_p$ , which is also a Sylow $p$-subgroup of $G=\mathbf{GL}_n(\mathbb{F}_p)$ .
I want to know that is there some fixed matrix $M_1\in U_1$ such that $NM_1N^{-1}\not\in U_1$ for all $N\in\mathbf{GL}_n(\mathbb{F}_p)-U$ , where $U$ is the group of all upper triangular matrices in $\mathbf{GL}_n(\mathbb{F}_p)$ .
If such matrix $M_1$ exists, I could show that the normalizer of $U_1$ in $G$ is contained in $U$ , i.e., $\mathbf{N}_{G}(U_1)\subseteq U$ . Conversely, we can easily check that $U\subseteq\mathbf{N}_{G}(U_1)$ . Then since $U_1\in\mathbf{Syl}_p(\mathbf{GL}_n(\mathbb{F}_p))$ , the equality $\mathbf{N}_{G}(U_1)=U$ implies that the number of Sylow $p$-subgroups of $G=\mathbf{GL}_n(\mathbb{F}_p)$ is $$ \begin{aligned} &|G:\mathbf{N}_{G}(U_1)| \\=&|G:U| \\=&\frac{|\mathbf{GL}_n(\mathbb{F}_p)|}{\left|\mathbb{F}_p^{\times}\right|^n\cdot|U_1|} \\=&\dfrac{\displaystyle\prod_{k=0}^{n-1}\left(p^n-p^k\right)}{(p-1)^n\cdot p^{n(n-1)/2}} \\=&\displaystyle\prod_{k=0}^{n-2}\left(1+\sum_{m=0}^{k}p^{m+1}\right)\ . \end{aligned} $$ I have tried to find some matrix $M_1$ which can be easily calculated by matrix multiplication and satisfies the properties as mentioned above, like the matrix defined as follows: $$ \begin{bmatrix} \ \ 1&0&\cdots&0&1\ \ \\&1&&&0\ \ \\&&\ddots&&\vdots\ \ \\&&&1&0\ \ \\&&&&1\ \ \end{bmatrix}\ , $$ but I am not sure that whether it is as desired.