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

In: The Theory of Nilpotent Groups

Author

Listed:
  • 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)

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.

Suggested Citation

  • Anthony E. Clement & Stephen Majewicz & Marcos Zyman, 2017. "“The Group Ring of a Class of Infinite Nilpotent Groups” by S. A. Jennings," Springer Books, in: The Theory of Nilpotent Groups, chapter 0, pages 219-268, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-66213-8_6
    DOI: 10.1007/978-3-319-66213-8_6
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