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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.

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  • $\begingroup$ Is $\;F=R=K\;$ ? I think there might be a little mess with the notation... $\endgroup$ – DonAntonio Jul 2 '16 at 10:48
  • $\begingroup$ @Joanpemo yes indeed, I changed it $\endgroup$ – user335236 Jul 2 '16 at 10:53
  • $\begingroup$ So in fact we have that $\;U(F)=F^*:=F\setminus\{0\}=\;$ the multiplicative group of the non-zero elements of the field $\;F\;$, and $\;U\le F^*\;$ ... all these are standard notations, I think, and $\;U(F)\;$ is usually reserved for the multiplicative group of units of a ring which is not a field or, at least, a division ring. $\endgroup$ – DonAntonio Jul 2 '16 at 10:57
  • $\begingroup$ I gather, also, that $\;F\;$ is a finite group, right? And if so, both $\;F^*\,,\,\,U\;$ are cyclic $\;p\,- $ groups. $\endgroup$ – DonAntonio Jul 2 '16 at 11:01
  • $\begingroup$ @Joanpemo yes of course, thanks for the comment, in an earlier version F was a Ring and I did'nt change the notation $\endgroup$ – user335236 Jul 2 '16 at 11:05
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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 :

$$P:=\left\{\;\;\begin{pmatrix}1&a_{12}&\ldots&a_{1n}\\0&1&\ldots&a_{2n}\\\ldots&\ldots&\ldots&\ldots\\0&0&\ldots&1\end{pmatrix}\in M_n(F)\;\;\right\}$$

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