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 D-0 = 1. Define Di+1 (G)/D-i(G) = D(G/D-i(G)) for i >= 1. By D-infinity(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-infinity(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. Published by Elsevier Inc.