On Residually Finite Groups of Finite General Rank

Following A. I. Maltsev, we say that a group $G$ has finite general rank if there is a positive integer $r$ such that every finite set of elements of $G$ is contained in some $r$-generated subgroup. Several known theorems concerning finitely generated residually finite groups are generalized here to the case of residually finite groups of finite general rank. For example, it is proved that the families of all finite homomorphic images of a residually finite group of finite general rank and of the quotient of the group by a nonidentity normal subgroup are different. Special cases of this result are a similar result of Moldavanskii on finitely generated residually finite groups and the following assertion: every residually finite group of finite general rank is Hopfian. This assertion generalizes a similar Maltsev result on the Hopf property of every finitely generated residually finite group.