9 problems

Let $G$ be a finite simple group, and let $H$ be a subgroup of $G$ whose index $[G:H]$ is odd. Vdovin–Revin conjecture. The subgroup $H$ is pronormal in $G$. This conjecture concer…

Let $H$ be a Hall subgroup of a finite group $G$. A subgroup $H$ is pronormal in $G$ if, for every $g\in G$, the subgroups $H$ and $H^g$ are conjugate in $\langle H,H^g\rangle$. It…

Normal-closure pronormality conjecture. The Hall subgroup $H$ is pronormal in its normal closure.

Pronormal-to-strongly-pronormal conjecture. Every pronormal Hall subgroup is strongly pronormal.

Sylow-tower strong pronormality conjecture. Every Hall subgroup having a Sylow tower is strongly pronormal.

Simple-group strong pronormality conjecture. Every Hall subgroup of a finite simple group is strongly pronormal.

Overgroup inheritance conjecture. If $H\le M\le G$, then $M\in D_\pi$.

Overgroup inheritance conjecture. If $H\le M\le G$, then $M\in C_\pi$.

Pronormality conjecture. Every subgroup of odd index in a finite simple group is pronormal.