Abstract

A group is said to have finite (special) rank <=s if all of its finitely generated subgroups can be generated by s elements. 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). © 1998 Academic Press.