A Non-hopfian Group by Gilbert Baumslag

The purpose of this note is to construct a non-hopfian and finitely generated group 5 which is in no way complicated. This group S is a generalised free square of the free nilpotent group A of class two on two generators. Since A satisfies the maximum condition, S differs radically from the non-hopfian groups constructed by Graham Higman [ l ] , with which it may be compared. I t may perhaps be of interest to point out that the first finitely generated non-hopfian groups were constructed by B. H. Neumann [2]. Besides these and the nonhopfian groups of Higman [ l ] , the only other known finitely generated non-hopfian groups are those due to B. H. Neumann and Hanna Neumann [3] and to P. Hall [4]. The construction of S is as follows. We take a replica B of A. Thus we may present A and B as follows: