Geodesic
Relative commutator associated with varieties of $n$-nilpotent and of $n$-solvable groups
Archivum mathematicum, Tome 42 (2006) no. 4, pp. 387-396.

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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.
Classification : 20E10, 20F12, 20F14
Keywords: relative commutator; nilpotent groups; solvable groups; central extensions 
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     author = {Everaert, Tomas and Gran, Marino},
     title = {Relative commutator associated with varieties of $n$-nilpotent and of $n$-solvable groups},
     journal = {Archivum mathematicum},
     pages = {387--396},
     publisher = {mathdoc},
     volume = {42},
     number = {4},
     year = {2006},
     mrnumber = {2283019},
     zbl = {1152.20030},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2006__42_4_a3/}
}
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Everaert, Tomas; Gran, Marino. Relative commutator associated with varieties of $n$-nilpotent and of $n$-solvable groups. Archivum mathematicum, Tome 42 (2006) no. 4, pp. 387-396. http://geodesic.mathdoc.fr/item/ARM_2006__42_4_a3/