A note on groups with a large permodularly embedded subgroup (Q6572417)

A famous theorem due to \textit{I. Schur} [Neuer Beweis eines Satzes uber endliche Gruppen, Berl. Ber., 1013--1019 (1902; JFM 33.0146.01)] ensures that if the center \(Z(G)\) of a group \(G\) has finite index, then the commutator subgroup \(G'\) is also finite. Since then the results which guarantee that a certain property of the factor group \(G/Z(G)\) is inherited by the subgroup \(G'\) are called of Schur type. For instance if the factor group \(G/Z(G)\) is polycyclic, then \(G'\) is also polycyclic.\N\NSchur's result can be equivalently formulated as follows: if a group \(G\) has a subgroup \(C\) of finite index such that \(\langle C, g\rangle\) is abelian for every \(g\in G\), then there exists a finite normal subgroup \(N\) of \(G\) such that the factor group \(G/N\) is abelian.\N\NIn [Arch. Math. 91, No. 2, 97--105 (2008; Zbl 1151.20021)], \textit{M. De Falco} et al. developed a lattice-theoretic approach to Schur's theorem and they prove that if a group \(G\) has a subgroup \(M\) of finite index such that \(\langle M,g\rangle\) has modular subgroup lattice for every \(g\in G\), then there exists a finite normal subgroup \(N\) of \(G\) such that the subgroup lattice of \(G/N\) is modular.\N\NA striking translation of normality in the subgroup lattice of a group is permodularity and from the same point of view, a good approximation of abelian groups are groups with a permodular subgroup lattice. Notice that the subgroup lattice of any quasi-Hamiltonian group is permodular.\N\NThe main result of this paper is the following:\N\NTheorem. Let \(G\) be a group if there exists a non-periodic subgroup \(P\) of \(G\) such that \(\langle P, g\rangle\) has permodular subgroup lattice for every \(g\in G\) (that is \(P\) is a permodularly embedded) and the interval \([G/P]\) is a polycyclic lattice, then \(G\) contains a polycyclic normal subgroup \(N\) such that \(G/N\) is quasi-Hamiltonian.\N\NAs a consequence of this theorem, a similar result can be proved when the interval \([G/P]\) is a supersoluble lattice. Moreover, the authors observe that in the main result the hypothesis that the subgroup \(P\) has elements of infinite order cannot be dropped.\N\NIn the final part of the paper, the structure of groups which are not polycyclic-by-quasi-Hamiltonian but have a permodularly embedded subgroup determining a polycyclic interval is described. Finally, the authors prove that is locally polycyclic every group \(G\) that contains a permodularly embedded subgroup \(P\) such that \([G/P]\) is a locally polycyclic lattice.