Introduction to Nilpotent Groups

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 2 in The Theory of Nilpotent Groups, 2017, pp 23-73 from Springer

Abstract: Abstract The aim of this chapter is to introduce the reader to the study of nilpotent groups. In Sect. 2.1, we define a nilpotent group, as well as the lower and upper central series of a group. Section 2.2 contains some classical examples of nilpotent groups. In particular, we prove that every finite p-group is nilpotent for a prime p. In Sect. 2.3, numerous properties of nilpotent groups are derived. For example, we prove that every subgroup of a nilpotent group is subnormal, and thus, satisfies the so-called normalizer condition. Section 2.4 is devoted to the characterization of finite nilpotent groups. In Sect. 2.5, we use tensor products to show that certain properties of a nilpotent group are inherited from its abelianization. We focus on torsion nilpotent groups in Sect. 2.6. We prove that every finitely generated torsion nilpotent group must be finite, and that the set of torsion elements of a nilpotent group form a subgroup. Section 2.7 deals with the upper central series and its factors. Among other things, we illustrate how the center of a group influences the structure of the group.

Date: 2017
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DOI: 10.1007/978-3-319-66213-8_2

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