Finding a $p$-Sylow-subgroup of GL$_2(\mathbb{Z}/p\mathbb{Z})$

Question

Can somebody help me to find a $p$-Sylow-subgroup of GL$_2(\mathbb{Z}/p\mathbb{Z})$?

I actually dont even know how to start :/

Thank you!

Answer 1

We have $|\text{GL}_2(\mathbb{Z}/p\mathbb{Z})|=p(p+1)(p-1)^2$.

Because the order of a $p$-Sylow-subgroup is the highest power of $p$ which divides the order of the group we get that every $p$-Sylow-subgroup has exactly $p$ elements.

Now let $A\in\text{GL}_2(\mathbb{Z}/p\mathbb{Z})$ with $A^p = E$, where $E$ denotes the identity matrix. From that we get that $|<A >|=p$.

So all thats left to do is to find a Matrix $A$ with $A^p =E$.

The Matrix $A=\begin{pmatrix}1&1\\0&1\end{pmatrix}$ does the job, because $\begin{pmatrix}1&1\\0&1\end{pmatrix}^p=\begin{pmatrix}1&0\\0&1\end{pmatrix} =E$.

Therefore $<A>$ is a $p$-Sylow-subgroup of $\text{GL}_2(\mathbb{Z}/p\mathbb{Z})$.