MAXIMAL SUBGROUPS OF SOME NON LOCALLY FINITE p-GROUPS.

Kaplansky's conjecture claims that the Jacobson radical $\mathcal{J}K[G]$ of a group algebra K[G], where K is a field of characteristic p > 0, coincides with its augmentation ideal $\mathcal{A}K[G]$ if and only if G is a locally finite p-group. By a theorem of Passman, if G is finitely generated and $\mathcal{J}K[G]=\mathcal{A}K[G]$ 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. [ABSTRACT FROM AUTHOR]