Locally Finite Groups with All Subgroups Normal-by-(Finite Rank)

A group is said to have finite (special) rank ≤ sif all of its finitely generated subgroups can be generated byselements. LetGbe a locally finite group and suppose thatH/H_Ghas finite rank for all subgroupsHofG, whereH_Gdenotes the normal core ofHinG. We prove that thenGhas an abelian normal subgroup whose quotient is of finite rank (Theorem 5). If, in addition, there is a finite numberrbounding all of the ranks ofH/H_G, thenGhas an abelian subgroup whose quotient is of finite rank bounded in terms ofronly (Theorem 4). These results are based on analogous theorems on locally finitep-groups, in which case the groupGis also abelian-by-finite (Theorems 2 and 3).