# nLab Hopfian group

Contents

### Context

#### Group Theory

group theory

Classical groups

Finite groups

Group schemes

Topological groups

Lie groups

Super-Lie groups

Higher groups

Cohomology and Extensions

Related concepts

# Contents

## Definition

### In sets

###### Definition

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.

### In concrete categories

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:

###### Definition

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.

###### Remark

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

###### Remark

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).

## Examples

###### Example

Clearly all finite groups are both Hopfian and coHopfian.

###### Example

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.

###### Example

Every torsion-free hyperbolic group? is Hopfian.

## References

• 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.