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

Abstract

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/HGhas finite rank for all subgroupsHofG, whereHGdenotes 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/HG, 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).