Upper-triangular matrices as semidirect product

Question

Let $p$ be a prime number and $G=GL_{n}(\mathbb{Z}/p\mathbb{Z})$. Consider the set $U_{n}$ of upper-triangular matrices of $G$ having entries of $1$ on the diagonal. The cardinality of $U_{n}$ is $p^{\frac{n(n-1)}{2}}$ and $U_{n} $ is a subgroup of $G$, \ in particular $U_{n}$ is a Sylow $p$-subgroup of $G $. Is it true that $U_{n}\simeq (\mathbb{Z}/p\mathbb{Z})^{n-1}\rtimes U_{n-1}.$

Answer 1

Yes. Indeed, the idempotent endomorphism described in block matrices as $p:\begin{pmatrix} A & b \\ 0 & 1 \end{pmatrix} \mapsto \begin{pmatrix}A& 0\\0&1\end{pmatrix}$ defines such a semidirect decomposition $\mathrm{Ker}(p)\rtimes\mathrm{Im}(p)$.