On the relationships between the factors of the upper and lower central series in some non-periodic groups

Abstract

This paper deals with the mutual relationships between the factor group $G/zeta(G)$ (respectively $G/zeta_k(G)$) and $G'$ (respectively $gamma_{k+1}(G)$ and $G^{mathfrak{N}}$). It is proved that if $G/zeta(G)$ (respectively $G/zeta_k(G)$) has finite $0$-rank, then $G'$ (respectively $gamma_{k+1}(G)$ and $G^{mathfrak{N}}$) also have finite $0$-rank. Furthermore, bounds for the $0$-ranks of $G', gamma_{k+1}(G)$ and $G^{mathfrak{N}}$ are obtained.

Keywords

• finite rank
• torsion-free rank
• section $p$-rank
• generalized radical group