On the intersections of solvable Hall subgroups in finite groups

Abstract

The following conjecture is considered: if a finite group G possesses a solvable π-Hall subgroup H, then there exist elements x, y, z, t ∈ G such that the identity H ∩ H x ∩ H y ∩ H z ∩ H t = O π(G) holds. Under additional conditions on G and H, it is shown that a minimal counterexample to this conjecture must be an almost simple group of Lie type.