Groups with complete lattice of nearly normal subgroups (Q1870097)

A subgroup \(H\) of a group \(G\) is said to be nearly normal in \(G\) if the index \(|H^G:H|\) is finite. The set \(\text{nn}(G)\) of all nearly normal subgroups is a sublattice of the subgroup lattice. However \(\text{nn}(G)\) is not always a complete lattice. The authors give some conditions when \(\text{nn}(G)\) is a complete lattice. The main result is Theorem 6. Let \(G\) be a group satisfying the following conditions: (i) \(G\) has a descending series of normal subgroups with finite factors; (ii) \(G\) has an ascending series of normal subgroups, every non-central factor of which is finite. If \(\text{nn}(G)\) is a complete lattice, then the derived subgroup of \(G\) is finite.