An Endomorphism of a Finitely Generated Residually Finite Group 
by Daniel T. Wise

Abstract:
Let $\phi:G\rightarrow G$ be an endomorphism of a finitely generated 
residually finite group. R.~Hirshon asked if there exists~$n$
such that the restriction of $\phi$ to $\phi^n(G)$ is injective. 
We give an example to show that this is not always the case.