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arXiv · 1611.01313

Narain Gupta's three normal subgroup problem and group homology

Roman Mikhailov·Inder Bir S. Passi
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Abstract

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.

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Roman Mikhailov, Inder Bir S. Passi. 2016-11-04. Narain Gupta's three normal subgroup problem and group homology. https://arxiv.org/abs/1611.01313

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