On Bouc's and Thévenaz's conjectures on B-groups

Bouc proposed the following conjecture: a finite group G is nilpotent if and only if its largest quotient B-group β(G) is nilpotent. And Thévenaz proposed another conjecture: a finite group G is solvable if and only if its largest quotient B-group β(G) is solvable. Bouc has proven that his conjecture holds when G is solvable. In this paper, we consider some cases when G is not solvable. Let S be a nonabelian simple group except the Chevalley groups An(q), Dn(q), E6(q), and An2(q), if the group G has exactly one composition factor isomorphic to S, then β(G) is not solvable, of course, is not nilpotent. That means we prove the conjectures in these cases.