On the norm of the nilpotent residuals of all subgroups of a finite group. (Q435944)

For a finite group \(G\) the authors define \(S(G)\) as the intersection of all normalizers of nilpotent residuals of subgroups. In iteration, put \(S_1(G)=S(G)\) and \(S_{n+1}(G)/S_n(G)=S(G/S_n(G))\), with \(S_\infty(G)\) as the last term.NEWLINENEWLINE The authors show: \(G=S_\infty(G)\) if and only if \(G\) is metanilpotent. Clearly \(G=S(G)\) if \(G\) is nilpotent or minimal nonnilpotent, but this class of groups is wider. -- If all elements of prime order of \(G\) are contained in \(S(G)\), then \(G\) is solvable, \(l_p(G)\leq 1\) for all odd primes, and \(G\) is of Fitting length at most \(3\). If also all elements of order \(4\) belong to \(S(G)\), then also \(l_2(G)\leq 1\) (Theorems 3.3, 5.2, 5.3).