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?

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 […]

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

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 ?

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

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, […]

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 $\varphi(G)$ […]

Intereting Posts

How find this integral $I=\int\frac{1}{\sin^5{x}+\cos^5{x}}dx$
consecutive prime power
Yitang Zhang: Prime Gaps
Determine whether a point lies inside the curve or outside a random curve using pencil and scale
Last few digits of $n^{n^{n^{\cdot^{\cdot^{\cdot^n}}}}}$
how to show $f$ attains a minimum?
Expected rank of a random binary matrix?
Good math bed-time stories for children?
Importance of Linear Algebra
Cyclic containment of sets
Number of non-decreasing functions between two finite sets
What contour should be used to evaluate $\int_0^\infty \frac{\sqrt{t}}{1+t^2} dt$
$\epsilon$-$\delta$ limit proof, $\lim_{x \to 2} \frac{x^{2}-2x+9}{x+1}$
An unusual combination lock problem
Straight Flush probability with a huge hand.