11 problems

Let $G$ be a finite group. For each pair of primes $p$ and $q$, suppose that $G$ contains a Hall subgroup for the set of primes $p,q\}$. P. Hall's solvability conjecture. Then $G$…

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…

Wielandt's conjecture. If $G$ satisfies both $mathscr D_sigma$ and $mathscr D_tau$, then $G$ satisfies $mathscr D_rho$.

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 $M\in C_\pi$.

Four-conjugate intersection conjecture. There exist elements $x,y,z,t\in G$ such that

$C_\pi$ inheritance conjecture. Then $A\in C_\pi$.