On the intersection of the normalizers of derived subgroups of all subgroups of a finite group

Given a finite group G, we define the subgroup D(G) to be the intersection of the normalizers of derived subgroups of all subgroups of G. Set D0=1. Define Di+1(G)/Di(G)=D(G/Di(G)) for i⩾1. By D∞(G) denote the terminal term of the ascending series. It is proved that the derived subgroup G′ is nilpotent if and only if G=D∞(G). Furthermore, if all elements of prime order of G are in D(G), then G is soluble with Fitting length at most 3. In Section 3, it is proved that if the group G satisfies G=D(G), then G′ is nilpotent and G″ has nilpotency class at most 2.