Welcome to mathimatikoi.org forum; Enjoy your visit here.

## Not a Hopfian group

Groups, Rings, Domains, Modules, etc, Galois theory
Riemann
Articles: 0
Posts: 169
Joined: Sat Nov 14, 2015 6:32 am
Location: Melbourne, Australia

### Not a Hopfian group

Definition: A group is called hopfian if every surjective homomorphism $f: \mathcal{G} \rightarrow \mathcal{G}$ is an isomorphism. Clearly every finite group is hopfian.

Problem:

Prove that

$\mathcal{G} = \langle x, y: y^{-1} x^2 y = x^3 \rangle$

is not hopfian.
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$