Narain Gupta's three normal subgroup problem and group homology

Roman Mikhailov, Inder Bir S. Passi

Published 2016-11-04Version 1

This paper is about application of various homological methods to classical problems in the theory of group rings. It is shown that the third homology of groups plays a key role in Narain Gupta's three normal subgroup problem. For a free group $F$ and its normal subgroups $R,\,S,\,T,$ and the corresponding ideals in the integral group ring $\mathbb Z[F]$, ${\bf r}=(R-1)\mathbb Z[F],\ {\bf s}=(S-1)\mathbb Z[F],\ {\bf t}=(T-1)\mathbb Z[F],$ a complete description of the normal subgroup $F\cap (1+{\bf rst})$ is given, provided $R\subseteq T$ and the third and the fourth homology groups of $R/R\cap S$ are torsion groups.