# proof of properties of Hopfian and co-Hopfian groups

. A group $G$ is Hopfian if and only if every surjective homomorphism $G\to G$ is an automorphism.

Proof. “$\Rightarrow$” Assume that $\psi:G\to G$ is a surjective homomorpism such that $\psi$ is not an automorphism, which means that $\mathrm{Ker}(\psi)$ is nontrivial. Then (due to the First Isomorphism Theorem) $G/\mathrm{Ker}(\psi)$ is isomorphic to $\mathrm{Im}(\psi)=G$. Contradiction, since $G$ is Hopfian.

$\Leftarrow$” Assume that $G$ is not Hopfian. Then there exists nontrivial normal subgroup $H$ of $G$ and an isomorphism $\phi:G/H\to G$. Let $\pi:G\to G/H$ be the quotient homomorphism. Then obviously $\pi\circ\phi:G\to G$ is a surjective homomorphism, but $\mathrm{Ker}(\pi\circ\phi)=H$ is nontrivial, therefore $\pi\circ\phi$ is not an automorphism. Contradiction. $\square$

Proposition. A group $G$ is co-Hopfian if and only if every injective homomorphism $G\to G$ is an automorphism.

Proof. “$\Rightarrow$” Assume that $\psi:G\to G$ is an injective homomorphism which is not an automorphism. Therefore $\mathrm{Im}(\psi)$ is a proper subgroup of $G$, therefore (since $\mathrm{Ker}(\psi)=\{e\}$ and due to the First Isomorphism Theorem) $G$ is isomorphic to its proper subgroup, namely $\mathrm{Im}(\psi)$. Contradiction, since $G$ is co-Hopfian.

$\Leftarrow$” Assume that $G$ is not co-Hopfian. Then there exists a proper subgroup $H$ of $G$ and an isomorphism $\phi:G\to H$. Let $i:H\to G$ be an inclusion homomorphism. Then $i\circ\phi:G\to G$ is an injective homomorphism which is not onto (because $i$ is not). Contradiction. $\square$

Title proof of properties of Hopfian and co-Hopfian groups ProofOfPropertiesOfHopfianAndCoHopfianGroups 2013-03-22 18:31:17 2013-03-22 18:31:17 joking (16130) joking (16130) 6 joking (16130) Proof msc 20F99