Let $F$ be a finite field of characteristic $p$ and $U$ a subgroup of $F^*$. Let $G$ be the Group of $n*n$ upper triangle matrices over $F$ with elements of $ U $ on the diagonal.
Find a Sylow-p-Group and a p-Complement.
I know that G is the semidirect Product of $T$ and $D$ with $T<G$ the subgroup of all unipotent elements in $G$ and $D<G$ the subgroup consisting of the diagonal matrices of $G$. Since $|G|=|T|*|D|$, I guess I will have to show that T is of order $p^k$ but for me this statements seems wrong.