normalizer quotient is $\operatorname{GL}_2(p)$

Question

Let $p$ be a prime and let $w$ be a primitive $p$-th root of unity in $\mathbb{C}$. There is an element $e$ of order $p$ in $G=\operatorname{PGL}_n(\mathbb{C})$ where $n=pk$ and $$e=\left[\left(\begin{matrix}I_k&&&&\\&{wI}_k&&&\\&&{w^2I}_k&&\\&&&\ddots&\\&&&&{w^{p-1}I}_k\\\end{matrix}\right)\right].$$ My focus is on the elementary abelian $p$-subgroups of rank 2 of $G$. Let $E$ be such a group.

Computation shows if all the nontrivial elements of $E$ are conjugate to $e$, then $N_{G}(E) /C_{G}(E) \cong \operatorname{GL}_2(p)$.

I can't see it clearly. I observe that if $E=\langle u,v \rangle = \langle uv,v \rangle = \langle u,uv \rangle$, then a change of basis matrix will be $$\left(\begin{matrix}1&0\\1&1\\\end{matrix}\right) \, \text{or} \, \left(\begin{matrix}1&1\\0&1\\\end{matrix}\right).$$

These two matrices generate $\operatorname{SL}_2(p)$. Am I arguing along the right lines?

Answer 1

Let $p$ be an odd prime. Look at the group $P$ generated by the following elements of $GL_p(\mathbb{C})$: $\left(\begin{smallmatrix} 1&&&\\&w&&\\&&\ddots&\\&&&w^{p-1}\end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}&1&&\\&&1&\\&&&\ddots\\1&&&&\end{smallmatrix}\right)$. Then $P$ is an extraspecial group of order $p^3$ and exponent $p$. Its image in $PGL_p(\mathbb{C})$ is elementary abelian $E$ of order $p^2$, and the normaliser modulo centraliser is $SL_2(p)$, acting as the group of all automorphisms of determinant one of $E$. There are $p-1$ representations like this, permuted by $GL_2(p)$; in other words, related by automorphisms of $E$. You can distinguish between these $p-1$ representations by the scalar representing a non-identity central element.

You can't mix different ones of these representations if the central element of $P$ has to act as a scalar on the entire space. So there are only two possibilities: Either your representation consists of $k$ copies of one of the $p-1$ representations above, in which case the normaliser modulo centraliser is $SL_2(p)$; or $k$ is divisible by $p$, and your representation is a direct sum of $p/k$ copies of the regular representation of the elementary abelian $E$, in which case the normaliser modulo centraliser is $GL_2(p)$.

Note I have edited this answer several times, for the sake of both correctness and readability. Apologies for any confusion caused.