The quasivariety generated by a torsion-free Abelian-by-finite group

Abstract

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 AB 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.

A quasivariety lattice of torsion-free soluble groups

“…It is proved that if G is a torsion-free finitely generated group in AB p k (here p is a prime, p = 2, and k ∈ N), which is a split extension of an Abelian group by a cyclic group, then the lattice L q (qG) is a finite chain. For p = 2, a similar result was derived in [7].…”