Let G be a group and C(G) denote the intersection of the normalizers of centralizers of all elements of G. Put C0(G) = 1 and define Ci+1(G)/Ci(G) = C(G/Ci(G)) for i " 0. Denote by C1(G) the terminal term of this ascending series. First we show, among other things, some basic properties of Cn(G) and C1(G) and then give a new characterization for nilpotent groups in terms of series in which defined via Cn(G).
