Chapter 2 Introduction to Nilpotent Groups The aim of this chapter is to introduce the reader to the study of nilpotent groups. In Section 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 Section 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 Section 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 Section 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. 2.1 The Lower and Upper Central Series In this section, we define a nilpotent group and discuss the lower and upper central series of a group. First, we provide some standard terminology. 2.1.1 Series of Subgroups Definition 2.1 Let G be a group. A series for G is a finite chain of subgroups 1 D G 0  G 1  G n D G: © Springer International Publishing AG 2017 A.E. Clement et al., The Theory of Nilpotent Groups, DOI 10.1007/978-3-319-66213-8_2 23