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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 7 in The Theory of Nilpotent Groups, 2017, pp 269-299 from Springer

Abstract: Abstract Chapter 7 contains a collection of miscellaneous topics. Section 7.1 pertains to M. Dehn’s algorithmic problems for finitely generated nilpotent group s. In Sect. 7.2, we prove that finitely generated nilpotent group s are Hopfian. Section 7.3 contains useful facts about groups of upper unitriangular matrices over a commutative ring with unity R. In Sect. 7.4, we study certain groups of automorphisms that are themselves nilpotent. In particular, we prove that if G is a nilpotent group of class c, then the the group of those automorphisms of G that induce the identity on the abelianization of G is nilpotent of class c − 1. Section 6.5 ends the chapter with an overview of the Frattini subgroup Φ(G) and Fitting subgroup Fit(G) of a group G. Among other results, we prove that if G is a finite group, then Φ(G) is nilpotent and Fit(G/Φ(G)) = Fit(G)/Φ(G) and Φ ( G ) ⊴ F i t ( G ) . $$\varPhi (G) \unlhd Fit(G).$$

Date: 2017
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DOI: 10.1007/978-3-319-66213-8_7

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