New posts in Hopfian Groups

Showing $\langle a,p,q\mid p^{-1}ap=a^2, q^{-1}aq=a^2\rangle$ is non-hopfian (from first principles).

Sort-of-simple non-Hopfian groups

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

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

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