# nLab Hopfian group

References

A discrete group $G$ is a Hopfian group if every surjective endomorphism $\phi : G\to G$ is an isomorphism. Dually, a discrete group is called coHopfian if any injective endomorphism of $G$ is an isomorphism.

As the epimorphisms and monomorphisms in Grp are precisely the surjections and injections (see epimorphisms of groups are surjective), the definition generalises immediately to that of a Hopfian object? in any category. In other words, we could define an object $X$ to be Hopfian if every epic endomorphism is an isomorphism.

Whatever that generalization is worth, much of the literature (such as V below) adopts the more concrete notion: given a concrete category $U: C \to Set$ with $U$ a faithful functor, say that an object $X$ of $C$ is Hopfian if every morphism $\phi: X \to X$ in $C$ with $U(\phi)$ surjective is an isomorphism. (In the presence of faithfulness of $U$, $\phi$ is epic if $U(\phi)$ is surjective.) For monadic functors $U: C \to Set$, this surjectivity assumption is the same as the assumption that $\phi$ is a regular epimorphism. Of course there is a dual notion of being co-Hopfian; here the hypothesis that $U(\phi)$ is injective frequently coincides simply with $\phi$ being monic – certainly that is true if $U$ preserves finite limits (which is frequently the case “in nature”).

Clearly all finite groups are both Hopfian and coHopfian. Using Nielsen’s method, one can show that every finitely generated free group and the union of any ascending chain of such free groups (for example, $\mathbb{Q}$) are Hopfian. It is also known that every torsion-free hyperbolic group? is Hopfian.

• K. Varadarajan, Hopfian and Co-Hopfian Objects, Publicacions Matemátiques, Vol. 36 (1992), 293-317. (web)