J. Group Theory 4 (2001), 325±340 Journal of Group Theory ( de Gruyter 2001 Groups with ®nite co-central rank Yaroslav Sysak and Achim Tresch (Communicated by F. de Giovanni) Abstract. A group G has ®nite co-central rank s if s is the least integer such that every ®nitely generated subgroup H can be generated by at most s elements modulo the centre of H. The relations between co-central rank and Pru È fer rank are studied. It is proved that the locally nilpotent torsion-free groups with ®nite co-central rank and the p-groups with ®nite co- central rank have ®nite Pru È fer rank modulo their centre. Each locally almost soluble group G with ®nite co-central rank is almost hyperabelian, and in the periodic case G has an abelian normal subgroup A such that G=A is residually ®nite and has ®nite Pru È fer rank. Furthermore, the locally graded periodic groups with ®nite co-central rank are locally ®nite. 1 Introduction Groups with ®nite Pru Èfer rank have been the subject of a large number of inves- tigations. Recall that a group G has ®nite Pru Èfer rank r  rG  if and only if every ®nitely generated subgroup can be generated by r elements, and r is minimal with this property. We introduce a less restrictive rank condition which does not immedi- ately a¨ect the structure of the abelian subgroups of a group. We shall say that a group G has ®nite co-central rank s  r c G  if for every ®nitely generated subgroup H of G, the group H can be generated by s elements modulo its centre ZH, and s is the least integer with this property. If no such integer exists, G has in®nite co-central rank. Obvious examples of groups with ®nite co-central rank are groups whose central factor groups have ®nite Pru È fer rank. Moreover, if G is a group with centre ZG  and if r  rG=ZG  is ®nite, then it is easy to see that r c G  W r. On the other hand, there exist groups of ®nite co-central rank whose central factor groups have in®nite rank and so the relation between these two ranks is more complicated than it seems at ®rst. For instance, it is not di½cult to verify that the wreath product C p o C p of two cyclic groups of prime order p has rank p and co-central rank 2 so that the direct product Dr p C p o C p , with p running through all primes, is a locally nilpotent group with this property. In the light of this example the following result establishes the best possible relation between Pru È fer rank and co-central rank. Theorem A. Let G be a locally nilpotent group with ®nite co-central rank s. Then G is hypercentral and the following statements hold. Brought to you by | University of Nebraska - Lincoln Authenticated Download Date | 11/5/14 7:40 PM