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Not a Hopfian group

Groups, Rings, Domains, Modules, etc, Galois theory
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Not a Hopfian group


Post by Riemann » Sun Jun 11, 2017 9:20 pm

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.


Prove that

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

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