Non-Hopfian groups with fully invariant kernels. I

Abstract:

Let $\mathcal {L}$ consist of the groups $G(l,m) = (a,b;{a^{ - 1}}{b^l}a = {b^m})$ where $|l| \ne 1 \ne |m|,lm \ne 0$ and $l,m$ are coprime. We characterize the endomorphisms of these groups, compute the centralizers of special elements and show that the endomorphism $a \to a,b \to {b^l}$ is onto with a nontrivial fully invariant kernel. Hence $G(l,m)$ is non-Hopfian in the’fully invariant sense.’