Hopfian group

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 $BS(2, 3)$.

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