On the norm of the centralizers of a group
Volume 149 / 2017
Colloquium Mathematicum 149 (2017), 87-91
MSC: Primary 20E34; Secondary 20F45.
DOI: 10.4064/cm6965-8-2016
Published online: 21 April 2017
Abstract
For any group $G$, let $C(G)$ denote the intersection of the normalizers of centralizers of all elements of $G$. Set $C_0= 1$. Define $C_{i+1}(G)/C_i(G)=C(G/C_i(G))$ for $i\geq 0$. Denote by $C_{\infty }(G)$ the terminal term of this ascending series. We show that a finitely generated group $G$ is nilpotent if and only if $G = C_{n}(G)$ for some positive integer $n$.
