RELATIVE COMMUTATOR ASSOCIATED WITH VARIETIES OF n-NILPOTENT AND OF n-SOLVABLE GROUPS
نویسنده
چکیده
A new notion of commutator was defined in any variety of Ω-groups A, relative to any fixed subvariety B of A [1]. This notion gives back the usual commutator of normal subgroups when A is the variety of groups and B the subvariety of abelian groups (see Section 2), but also the Peiffer commutator of normal precrossed submodules, when A is the variety of precrossed modules and B the subvariety of crossed modules. It is the aim of this paper to investigate this new notion of relative commutator in the variety of groups with respect to the classical subvarieties of n-nilpotent and of n-solvable groups. Although the characterization of the relative commutator is not trivial in these specific examples, it is fortunate that it can be expressed by a simple formula in terms of the usual commutator of normal subgroups. For instance, when A = Gp, the variety of groups, and B = Nil2, the subvariety of nilpotent groups of class at most 2, the relative commutator [M, N ]Nil2 of normal subgroups M and N of a group A is given by [M, N ]Nil2 = [
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