Isolators, Extraction of Roots, and P-Localization

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 5 in The Theory of Nilpotent Groups, 2017, pp 155-217 from Springer

Abstract: Abstract When considering rings acting on groups, one may specialize to subrings of ℚ . $$\mathbb {Q}.$$ A group action by a subring of ℚ $$\mathbb {Q}$$ may be interpreted as root extraction in the said group, which is the underlying subject of Chap. 5. In Sect. 5.1, we discuss the theory of isolators and isolated subgroups. This topic is related to the study of root extraction in nilpotent groups, which is the theme of Sect. 5.2. We discuss some of the theory of root extraction in groups, emphasizing the main results for nilpotent groups. Residual properties of nilpotent groups are also discussed in Sect. 5.2, as well as embeddings of nilpotent groups into nilpotent groups with roots. Section 5.3 contains an introduction to the theory of localization of nilpotent groups. A group is said to be P-local, for a set of primes P, if every group element has a unique nth root whenever n is relatively prime to the elements of P. The main result in this section is: Given a nilpotent group G, there exists a nilpotent group G P of class at most the class of G which is the best approximation to G among all P-local nilpotent groups.

Date: 2017
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DOI: 10.1007/978-3-319-66213-8_5

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