Maximal quasivarieties of groups
Copyright policy )Maximal quasivarieties of groups
Lattices \(L_q(qG)\) of quasivarieties contained in the quasivariety \(qG\), generated by a polycyclic-by-finite group \(G\), are considered. It is proved that every lattice under consideration contains finitely many coatoms (i.e. proper maximal elements) and that each of its elements distinct from \(qG\) is contained in some coatom. Earlier, the author and \textit{V. A. Gorbunov} [Algebra Logika 14, 123-142 (1975; Zbl 0317.08003)] proved the same for the case in which \(G\) is a finite group. The author also constructs an example of a finitely generated solvable group \(B\) of derived length 3 such that there are no coatoms in the quasivariety lattice \(L_q(qB)\).
Generalizations of solvable and nilpotent groups, Solvable groups, supersolvable groups, polycyclic-by-finite groups, finitely generated groups, Quasivarieties and varieties of groups, Lattices of varieties, quasivarieties of groups, quasivariety lattices, maximal quasivarieties, Quasivarieties, coatoms, solvable groups
Generalizations of solvable and nilpotent groups, Solvable groups, supersolvable groups, polycyclic-by-finite groups, finitely generated groups, Quasivarieties and varieties of groups, Lattices of varieties, quasivarieties of groups, quasivariety lattices, maximal quasivarieties, Quasivarieties, coatoms, solvable groups
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