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.
So we have that
$$G:=\left\{\;\;\begin{pmatrix}a_{11}&a_{12}&\ldots&a_{1n}\\0&a_{22}&\ldots&a_{2n}\\\ldots&\ldots&\ldots&\ldots\\0&0&\ldots&a_{nn}\end{pmatrix}\in M_n(F)\;/\;\;a_{ii}\in U\;\;\right\}$$
The order of the above group is, if $\;F=\Bbb F_{p^k}\;$ :
$$(p^k-1)^n\cdot\left(p^k\right)^{(n-1)+(n-2)+\ldots+2+1}=(p^k-1)^np^{\frac{(n-1)n}2k}$$
Thus, I'd propose the following as $\;p\,-$ subgroup of $\;G\;$ if you need the express form, and with what you did and Derek's comment you've almost finished, I think :
$$P:=\left\{\;\;\begin{pmatrix}1&a_{12}&\ldots&a_{1n}\\0&1&\ldots&a_{2n}\\\ldots&\ldots&\ldots&\ldots\\0&0&\ldots&1\end{pmatrix}\in M_n(F)\;\;\right\}$$