$U_n(\mathbb{F}_p)$ is the group of upper triangular matrices of order n with diagonal entries 1 and other entries from $\mathbb{F}_p$ (equipped with matrix multiplication). $$U_n = \left \{\left ( \begin{matrix} 1 & &* \\ & 1 & \\ 0& & 1 \end{matrix}\right )_n : * \in \mathbb{F}_p \right \}$$
I need to show that given any p-group $G$, it is isomorphic to a subgroup of $U_n$ where $|G| = n$
One of the hints that I have been given is to see that there is an element in $\mathbb{F}_p^n$ which is stabilised by all elements of $G$ when they are viewed as members of $GL_n(\mathbb{F}_p)$. I have proved this but I do not know how to proceed from here.
Let $\rho$ be the regular representation for the $p$-group $G$. We have that kernel of $\rho$ is trivial so $\rho$ is an injection of $G$ into $GL_n(p)$. Identify $G$ with its image. As $G$ is a $p$-subgroup of $GL_n(p)$, it is a subgroup of some Sylow $p$-subgroup of $GL_n(p)$. Note that $UT_n(p)$ is a Sylow $p$-subgroup of $GL_n(p)$ and all Sylow $p$-subgroups are conjugate, and so they are isomorphic. Therefore, we can identify $G$ with a subgroup of $UT_n(p)$.