2 problems

Bouc's conjecture. $\beta(G)$ is nilpotent if and only if $G$ is nilpotent.

Let $G$ be a finite group, and let $\beta(G)$ denote its largest quotient that is a $B$-group. Thévenaz's conjecture. The group $\beta(G)$ is solvable if and only if $G$ is solvabl…