Let $H$ be a subgroup of $G$. Then a homomorphism $r : G \to H$ is said to be a retraction if $r(x) = x$ for all elements $x\in H$. Then $H$ is called a retract of $G$.
A nontrivial group is said to be splitting-simple if it has no proper nontrivial retracts. Clearly every simple group is splitting-simple. Also $\mathbb{Z}$ is a non-simple splitting-simple group.
It can be easily seen that every simple group is Hopfian (recall that a group $G$ is Hopfian if every epimorphism $f : G \to G$ is an automorphism).
My question: Is every splitting-simple group $G$ Hopfian? If not, what is the necessity for $G$ to be Hopfian?
The answer to the first question is. No.
The Prüfer $p$-group is non-Hopfian group, but it has no proper nontrivial retracts, since each of its subgroups is finite and any homomorphic image is a divisible group.