“The Group Ring of a Class of Infinite Nilpotent Groups” by S. A. Jennings

Anthony E. Clement, Stephen Majewicz and Marcos Zyman
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Anthony E. Clement: CUNY-Brooklyn College, Department of Mathematics
Stephen Majewicz: CUNY-Kingsborough Community College, Mathematics and Computer Science
Marcos Zyman: CUNY-Borough of Manhattan Community College, Department of Mathematics

Chapter Chapter 6 in The Theory of Nilpotent Groups, 2017, pp 219-268 from Springer

Abstract: Abstract This chapter is based on a seminal paper entitled “The Group Ring of a Class of Infinite Nilpotent Groups” by S. A. Jennings. In Sect. 6.1, we consider the group ring of a finitely generated torsion-free nilpotent group over a field of characteristic zero. We prove that its augmentation ideal is residually nilpotent. We introduce the dimension subgroups of a group in Sect. 6.2. These subgroups are defined in terms of the augmentation ideal of the corresponding group ring. We prove that the nth dimension subgroup coincides with the isolator of the nth lower central subgroup. This is a major result involving a succession of clever reductions where nilpotent groups play a prominent role. Section 6.3 deals with nilpotent Lie algebras. We show that there exists a nilpotent Lie algebra over a field of characteristic zero which is associated to a finitely generated torsion-free nilpotent group. As it turns out, the underlying vector space of this Lie algebra has dimension equal to the Hirsch length of the given group.

Date: 2017
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DOI: 10.1007/978-3-319-66213-8_6

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