 Non-Hopfian group

A non-Hopfian group admits a surjective endomorphism which is not injective. The Baumslag-Solitar group $BS(2,3)$ has such an endomorphism $a\mapsto a^2$, $t\mapsto t$: its image contains $a^2$ and $a^3$, while the nonidentity commutator $[a,ta{t}^{-1}]$ lies in its kernel by Britton's lemma.