Relative commutator associated with varieties of $n$-nilpotent and of $n$-solvable groups

T. Everaert, M. Gran

Address. Vakgroep Wiskunde, Faculteit Wetenschappen, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium
Lab. Math. Pures et Appliqu\'ees, Universit\'e du Littoral C\^ote d'Opale, 50, Rue F. Buisson, 62228 Calais, France


Abstract. In this note we determine explicit formulas for the relative commutator of groups with respect to the subvarieties of $n$-nilpotent groups and of $n$-solvable groups. In particular these formulas give a characterization of the extensions of groups that are central relatively to these subvarieties.

AMSclassification. Primary: 20F12, 20F16, 20F18; Secondary: 08B99, 20K35.

Keywords. Relative commutator, nilpotent groups, solvable groups, central extensions.