Bouc's conjecture on $B$-groups

Abstract

Bouc proposed the following conjecture: a finite group $G$ is nilpotent if and only if its largest quotient $B$-group $\beta(G)$ is nilpotent. And he has prove that this conjecture holds when $G$ is solvable. In this paper, we consider the case when $G$ is not solvable. Let $S$ be a nonabelian simple group except the Chevalley groups $A_{n}(q)$, $D_{n}(q)$, $E_{6}(q)$, and $^2A_{n}(q)$, if there exists only one factor of $G$ which is isomorphic to $S$, then $\beta(G)$ is not solvable, of course, is not nilpotent. That means we prove the conjecture in these cases.