6 problems

Let $G$ and $C$ be ranked connected groups such that $C<G$, $[N_G(C):C]<\finite$, and $C\cap C^g=\{1\}$ for every $g\notin N_G(C)$. Suppose that $C<G$ is a ranked quasi-Frobenius g…

Let $G$ be a simple ranked group, and let $C$ be a Cartan subgroup, that is, the centraliser of a maximal decent torus. An element is strongly real if it is a product of two involu…

Let $G$ be a simple ranked group, and let a Borel subgroup mean a definable, connected, soluble subgroup maximal among such subgroups. Nilpotent Borel subgroup conjecture. If all B…

Let $G$) be a connected ranked group with involutions but with no infinite elementary abelian subgroup. Suppose that $G$ has a definable, connected subgroup $C<G$ which is a TI-sub…

Fixed-point implication conjecture. If the group of fixed points of $\alpha$ is pseudofinite, then $G$ is isomorphic to a Chevalley group over an algebraically closed field.

Hrushovski's fixed-point conjecture. The group of fixed points of $\alpha$ is pseudofinite, that is, an infinite model of the theory of finite groups.