Groups with finitely many isomorphism classes of non-modular subgroups (Q6639814)

A subgroup \(H\) of a group \(G\) is modular if it is a modular element of the lattice \(\mathcal{L}(G)\) of all subgroups of \(G\). In other words, this means that \(\langle X, H \cap Y \rangle = \langle X, H \rangle \cap Y \) for all \(X,Y \leq G\) such that \(X \leq Y\) and \(\langle H, X \cap Y\rangle =\langle H, X \rangle \cap Y\) if \(H \leq Y\).\N\NThe main results in the paper under review are the following.\N\NTheorem A: Let \(G\) be an infinite locally finite group. Then the set of all non-modular subgroups of \(G\) has finite isomorphism type if and only if either \(G\) has modular subgroup lattice or \(G\) contains a central Prüfer subgroup \(P\) and a finite subgroup \(E\) such that \(G=PE\) and \(G/(P\cap E)\) has modular subgroup lattice.\N\NTheorem B. Let \(G\) be a locally generalized radical group for groups in which the set of all non-permutable subgroups has a finite isomorphism type. If \(G\) does not have a modular subgroup lattice, then \(G\) is a soluble-by-finite minimax group whose finite residual is either trivial or a Prüfer group.\N\NMoreover, the author obtains the analogues of Theorems A and B for groups in which the set of all non-permutable subgroups has finite isomorphism type (see Theorems C and D).