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Commutator Calculus

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

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.

Suggested Citation

  • Anthony E. Clement & Stephen Majewicz & Marcos Zyman, 2017. "Commutator Calculus," Springer Books, in: The Theory of Nilpotent Groups, chapter 0, pages 1-21, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-66213-8_1
    DOI: 10.1007/978-3-319-66213-8_1
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