- group, ∞-group
- group object, group object in an (∞,1)-category
- abelian group, spectrum
- super abelian group
- group action, ∞-action
- representation, ∞-representation
- progroup
- homogeneous space

**Classical groups**

**Finite groups**

**Group schemes**

**Topological groups**

**Lie groups**

**Super-Lie groups**

**Higher groups**

**Cohomology and Extensions**

**Related concepts**

A discrete group $G$ is called a **Hopfian group** if every surjective endomorphism $\phi \colon G\to G$ is an isomorphism.

Dually, it 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 epimorphic endomorphism is an isomorphism (cf. *Dedekind finite object*).

Whatever that generalization is worth, much of the literature (such as Varadarajan 1992) adopts the more concrete notion:

Given a concrete category $U \colon C \to Set$ with $U$ a faithful functor, say that an object $X$ of $C$ is *Hopfian* if every morphism $\phi \colon X \to X$ in $C$ with $U(\phi)$ surjective is an isomorphism.

Notice that if $U$ if faithful, then $\phi$ is epimorpic if $U(\phi)$ is surjective.

For monadic functors $U \colon C \to Set$, 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 monomorphic – certainly that is true if $U$ preserves finite limits (which is frequently the case in application).

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.

Every torsion-free hyperbolic group? is Hopfian.

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

Last revised on January 22, 2024 at 16:21:49. See the history of this page for a list of all contributions to it.