Locally finite groups with a nilpotent or locally nilpotent maximal subgroup

We prove that a locally finite group G containing a core-free, locally nilp otent FC-subgroup M, which is maximal in G, is locally solvable if M is not a 2-group. We also prove that the open question of solvability of a locall y finite group G containing a, core-free, nilpotent p-subgroup M, which is maximal in G has a positive answer if p greater than or equal to 5 or if p = 3 and M is metabelian.