Groups which are isomorphic to their nonabelian subgroups
Let \(\mathcal X\) be the class of all groups \(G\) containing proper non-abelian subgroups and such that all these subgroups are isomorphic to \(G\). The authors give a complete description of soluble \(\mathcal X\)-groups, and prove that if \(G\) is an unsoluble \(\mathcal X\)-group, then \(G/Z(G)\) is simple and each non-abelian subgroup of \(G\) contains \(Z(G)\). It follows in particular that every locally graded \(\mathcal X\)-group is soluble (a group \(G\) is locally graded if every finitely generated non-trivial subgroup of \(G\) contains a proper subgroup of finite index). The problem of the existence of unsoluble \(\mathcal X\)-groups is left here as an open question.
scientific article; zbMATH DE number 1089765