On intersections of abelian and nilpotent subgroups in finite groups. I

Abstract

Let A be an abelian subgroup of a finite group G, and let B be a nilpotent subgroup of G. If G is solvable, then we prove that it contains an element g such that A ∩ B^g ≤ F(G), where F(G) is the Fitting subgroup of G. If G is not solvable, we prove that a counterexample of minimal order to the conjecture that A ∩ B^g ≤ F(G) for some element g from G is an almost simple group.