This question popped up as an exercise in a mathematics magazine. Thoughts are that we're looking at an infinite group but beyond that I'm stumped. Any ideas/examples would be fantastic!
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2For instance, $\Bbb C^{} / {±1} \cong \Bbb C^{}$. It has already been asked many times on M.SE. – Watson Sep 25 '16 at 16:21
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2Precisely when the group is not Hopfian. – H.Durham Sep 25 '16 at 16:52
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Apropriate answers are already given in the comments so let me just summarize them and add only a little bit.
1) Clearly such a group must be infinite (for a finite group this cannot be true by a simple cardinality argument).
2) $\mathbb{C}^*$ is an example. (comment of @Watson)
3) $\mathbb{Z}^\mathbb{N}$ is an example. Or more generally $G^\mathbb{N}$ for any non-trivial group $G$.
4) $BS(2,3)$ is a finitely generated example. (Baumslag Solitar group)
5) Such groups are called non-hopfian. (comment of @H.Durham)
6) As far as I know they are called hopfian/non-hopfian since the famous mathematician Hopf asked if such groups exists which are finitely generated.
7) There is a famous theorem of Mal'cev which states that a finitely generated residually finite group is Hopfian.
M.U.
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Not a problem. Incidentally, there are hopfian BS groups which are not solvable. – Moishe Kohan Sep 26 '16 at 13:28