On the norm of the centralizers of a group

Mohammad Zarrin¹

¹ Department of Mathematics University of Kurdistan P.O. Box 416, Sanandaj, Iran

Colloquium Mathematicum, Tome 149 (2017) no. 1, pp. 87-91.

Résumé

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 $n$.

Détail

DOI : 10.4064/cm6965-8-2016

Keywords: group denote intersection normalizers centralizers elements set define i geq denote infty terminal term ascending series finitely generated group nilpotent only positive integer