On a question concerning the intersection of Φ-isolators of finite soluble groups

Abstract

A subgroup A of a finite group G is called Φ-isolator of G if A covers the Frattini chief factors of G and avoids the supplemented ones. Let H be a Φ-isolator of a soluble finite group G. In the present paper, we prove that there exist elements x,y∈G such that the equality H∩Hx∩Hy=Φ(G) holds. This result is an answer to Question 19.38 in “The Kourovka notebook”.