On the norm of the nilpotent residuals of all subgroups of a finite group

R. Baer and Wielandt in 1934 and 1958, respectively, considered the intersection of the normalizers of all subgroups of G and the intersection of the normalizers of all subnormal subgroups of G. In this paper, for a finite group G, we define the subgroup S(G) to be the intersection of the normalizers of the nilpotent residuals of all subgroups of G. Set S0=1. Define Si+1(G)/Si(G)=S(G/Si(G)) for i⩾1. By S∞(G) denote the terminal term of the ascending series. It is proved that G=S∞(G) if and only if the nilpotent residual GN is nilpotent. Furthermore, if all elements of prime order of G are in S(G), then G is solvable and lp(G)⩽1, where lp(G) is p-length of G for p∈π(G), where π(G) means the set of prime divisors of |G|.