Commutator Calculus

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 1 in The Theory of Nilpotent Groups, 2017, pp 1-21 from Springer

Abstract: Abstract In this chapter, we introduce the commutator calculus. This is one of the most important tools for studying nilpotent groups. In Sect. 1.1, the center of a group and other notions surrounding the concept of commutativity are defined. Several results and examples involving central subgroups and central elements are given. Section 1.2 contains the fundamental identities related to commutators of group elements. By definition, the commutator of two elements g and h in a group G is the element [g, h] = g −1 h −1 gh. Clearly, [g, h] = 1 whenever g and h commute. This leads to a natural connection between central elements and trivial commutators. The commutator identities allow us to develop properties of commutator subgroups. This is the main focus of Sect. 1.3.

Date: 2017
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DOI: 10.1007/978-3-319-66213-8_1

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