LOCALLY FINITE-GROUPS WITH ALL SUBGROUPS NORMAL-BY-(FINITE RANK)

A group is said to have finite (special) rank less than or equal to s if all of its finitely generated subgroups can be generated by s eleme nts. Let G be a locally finite group and suppose that H/H-G has finite rank for all subgroups H of G, where H-G denotes the normal core of H in G. We prove that then G has an abelian normal subgroup whose quoti ent is of finite rank (Theorem 5). If, in addition, there is a finite number r bounding all of the ranks of H/H-G, then G has an abelian sub group whose quotient is of finite rank bounded in terms of r only (The orem 4). These results are based on analogous theorems on locally fini te p-groups, in which case the group G is also abelian-by-finite (Theo rems 2 and 3). (C) 1998 Academic Press.