On the Derived Norm of a Finite Group

Given a finite group G , we define a subgroup CS ( G ) as the intersection of the normalizers of all subgroups of the derived subgroup of G . Let CS 0 = 1. We define CS i +1 ( G )/ CS i ( G ) = CS ( G / CS i ( G )) for i ≥ 1. By CS ∞ ( G ) we denote the terminal term of the upper series. It is proved that the derived subgroup G ′ is nilpotent if and only if G = CS ∞ ( G ). In particular, we obtain the following result: if all elements of odd prime order of G are in CS ( G ), then G is solvable.