Narain Gupta's three normal subgroup problem and group homology

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. Fo r a free group Fand its normal subgroups R, S, T, and the corresponding ideals in the integral group ring Z[F], r =(R−1)Z[F], s =(S−1)Z[F], t =(T−1)Z[F], acomplete description of the normal subgroup F∩(1 +rst)is given, provided R⊆Tand the third, fourth and fifth homology groups of R/R∩Sare torsion groups.