Geodesic
The quasivariety generated by a~torsion-free Abelian-by-finite group
Algebra i logika, Tome 46 (2007) no. 4, pp. 407-427.

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Let $L_q(qG)$ be the quasivariety lattice contained in a quasivariety generated by a group $G$. It is proved that if $G$ is a finitely generated torsion-free group in $\mathcal A\mathcal B_{2^n}$ (i.e., $G$ is an extension of an Abelian group by a group of exponent $2^n$), which is a split extension of an Abelian group by a cyclic group, then the lattice $L_q(qG)$ is a finite chain.
Keywords: quasivariety, quasivariety lattice, metabelian group. 
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A. I. Budkin. The quasivariety generated by a~torsion-free Abelian-by-finite group. Algebra i logika, Tome 46 (2007) no. 4, pp. 407-427. http://geodesic.mathdoc.fr/item/AL_2007_46_4_a0/

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