A note on groups with a large permodularly embedded subgroup

Abstract

AbstractIt is known that if G is a group such that the centre factor group $$G/\zeta (G)$$ G / ζ ( G ) is polycyclic, then also the commutator subgroup $$G'$$ G ′ is polycyclic. The aim of this paper is to describe this situation from a lattice point of view. It is proved that if G is a group admitting a permodularly embedded non-periodic subgroup P such that the interval [G/P] is a polycyclic lattice, then G contains a polycyclic normal subgroup N such that G/N is quasihamiltonian.