Nonsurjective epimorphisms in decomposable varieties of groups

Abstract

A full characterization of when a subgroup $H$ of a group $G$ in a varietal product $\mathcal{N}\mathcal{Q}$ is epimorphically embedded in $G$ (in the variety $\mathcal{N}\mathcal{Q}$) is given. From this, a result of S.~McKay is derived, which states that if $\mathcal{N}\mathcal{Q}$ has instances of nonsurjective epimorphisms, then $\mathcal{N}$ also has instances of nonsurjective epimorphisms. Two partial converses to McKay's result are also given: when~$G$ is a finite nonabelian simple group; and when~$G$ is finite and $\mathcal{Q}$ is a product of varieties of nilpotent groups, each of which contains the infinite cyclic group.

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References in corpus (1)

• Dominions in the variety of metabelian groups