Relative commutator associated with varieties of $$n$$-nilpotent and of $$n$$-solvable groups.(English)Zbl 1152.20030

The notion of commutator of normal subgroups relative to any given variety of groups was introduced recently by T. Everaert [J. Pure Appl. Algebra 210, No. 1, 1-10 (2007; Zbl 1117.08007)]. In this paper, explicit formulas expressing commutators with respect to the varieties of $$n$$-nilpotent and $$n$$-solvable groups by means of standard commutators are given. For the case of $$n=2$$, detailed proofs are provided. These formulas in particular give characterizations of the extensions of groups that are central (in the sense of A. Fröhlich [Trans. Am. Math. Soc. 109, 221-244 (1963; Zbl 0122.25702)]) with respect to these varieties.
Reviewer: Michal Kunc (Brno)

MSC:

 20F12 Commutator calculus 20E10 Quasivarieties and varieties of groups 20F16 Solvable groups, supersolvable groups 20F18 Nilpotent groups

Citations:

Zbl 1117.08007; Zbl 0122.25702
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