Articles of hopfian groups

The Hopfian property for groups

Let $G$ be a group, which for my purposes would be abelian. To say that $G$ has the Hopf property is to say that every epimorphism of $G$ is an automorphism. Does anyone happen to recall the context in which Hopf first used this concept, and a reference for this?

Can you always find a surjective endomorphism of groups such that it is not injective?

If we take the following endomorphism, $\phi:R[t] \to R[t]$ by $\sum_{i = 0}^n a_it^i \mapsto \sum_{i = 0}^{\lfloor n/2 \rfloor} a_{2i} t^i$, it is surjective but not injective. (It just removes odd coefficients and pushes everything down). Is there a similar endomorphism from $\mathbb{Z} \to \mathbb{Z}$? If so, can you give an example. Otherwise what […]

If $G$ is a group and $N$ is a nontrivial normal subgroup, can $G/N \cong G$?

This question already has an answer here: Does $G\cong G/H$ imply that $H$ is trivial? 10 answers

Does $G\cong G/H$ imply that $H$ is trivial?

Let $G$ be any group such that $$G\cong G/H$$ where $H$ is a normal subgroup of $G$. If $G$ is finite, then $H$ is the trivial subgroup $\{e\}$. Does the result still hold when $G$ is infinite ? In what kind of group could I search for a counterexample ?

An example of a residually finite group which is not Hopf

trying to think of any residually finite group which is not Hopf. Any help?

Sort-of-simple non-Hopfian groups

A finite simple group is one which has no homomorphic images apart from itself and the trivial group. However, the simple-groups tag does not include the condition “finite”. My question is the following. Is the following true? Claim: A simple group is one where the only homomorphic images are itself and the trivial group. However, […]

When can a pair of groups be embedded in each other?

This is a question I made up, but couldn’t solve even after some days’ thought. Also if any terminology is unclear or nonstandard, please complain. Given groups $G$ and $H$, we say that $G$ can be embedded in $H$ if there exists an injective homomorphism $\varphi : G \to H$. (Note that the image $\varphi(G)$ […]