Maximal Subgroups of some Non Locally Finite P-groups

Kaplansky's conjecture claims that the Jacobson radical of a group algebra K[G], where K is a field of characteristic p > 0, coincides with its augmentation ideal if and only if G is a locally finite p-group. By a theorem of Passman, if G is finitely generated and then any maximal subgroup of G is normal of index p. In the present paper, we consider one infinite series of finitely generated infinite p-groups (hence not locally finite p-groups), so called GGS-groups. We prove that their maximal subgroups are nonetheless normal of index p. Thus these groups remain among potential counterexamples to Kaplansky's conjecture.