---
tag: 6S4CM98021
kind: definition
name: Hopfian group
msc: ["20E26", "20E36"]
defines: PHEN4E1JPT
uses: [G5S1M9, 0JYTG5K9Y4, 01RVCMSN5N, 09NWETVTNY]
sources:
  - url: https://en.wikipedia.org/wiki/Hopfian_group
---

## Definition

Let $G$ be a [[group|G5S1M9]].
$G$ is *Hopfian* if every [[surjective|01RVCMSN5N]] group [[homomorphism|0JYTG5K9Y4]] $\varphi \colon G \to G$ is [[injective|09NWETVTNY]].

## Remarks

Equivalently, if $N$ is a normal subgroup of $G$ and $G/N \cong G$, then $N = \set{1}$.

Example: $\Z$ is Hopfian.
A homomorphism $\varphi \colon \Z \to \Z$ is $n \mapsto kn$ with $k = \varphi(1)$.
If it is surjective, then $k = \pm 1$, and then it is injective.

Non-example: let $G = \bigoplus_{n \ge 1} \Z/2$, the group of sequences $(x_1, x_2, \ldots)$ in $\Z/2$ with only finitely many nonzero terms.
The shift $\varphi(x_1, x_2, x_3, \ldots) = (x_2, x_3, \ldots)$ is a surjective homomorphism.
It is not injective, since $\varphi(1, 0, 0, \ldots) = 0$.
A finitely generated non-Hopfian group is the [[Baumslag–Solitar group|429JJ5ZNBY]] $BS(2, 3)$.

The dual notion is a co-Hopfian group: every injective homomorphism from the group to itself is surjective.