Centralizers and norm of a group

‎Let $G$ be a group and $C(G)$ denote the intersection of the normalizers of‎ ‎centralizers of all elements of $G$‎. ‎Put $C_0(G)=1$ and define $C_{i+1} (G)/C_{i} (G) = C (G/C_{i} (G))$ for $i \ge 0$‎. ‎Denote by $C_\infty(G)$ the terminal term of this ascending series‎. ‎First we show‎, ‎among other things‎, ‎some basic properties of $C_n(G)$ and $C_{\infty}(G)$ and then give a new characterization for nilpotent groups in terms of series in which defined via $C_{n}(G)$