17.42 (2010)

Let $\overline{G}$ be a simple algebraic group of adjoint type over the algebraic closure $\mathbb{F}_p$ of a finite field $\mathbb{F}_p$ of prime order $p$, and $\sigma$ a Frobenius map (that is, a surjective homomorphism such that $G_\sigma = C_{\overline{G}}(\sigma)$ is finite). Then $G = O^{p'}(G_\sigma)$ is a finite group of Lie type. For a maximal $\sigma$-stable torus $T$ of $\overline{G}$, let $N = N_{\overline{G}}(T) \cap G$. Assume also that $G$ is simple and $G \not\cong \text{SL}_3(2)$. Does there always exist $x \in G$ such that $N \cap N^x$ is a $p$-group?