# When is a group isomorphic to a non-trivial quotient group of itself? [duplicate]

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!

• For instance, $\Bbb C^{*} / \{±1\} \cong \Bbb C^{*}$. It has already been asked many times on M.SE. – Watson Sep 25 '16 at 16:21
• Precisely when the group is not Hopfian. – H.Durham Sep 25 '16 at 16:52

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.

• Not BS(1,2) (that one is Hopfian) but BS(2,3), say. – Moishe Kohan Sep 25 '16 at 22:44
• upps, I feel ashamed now ... you are absolutely right :) – M.U. Sep 26 '16 at 6:59
• Not a problem. Incidentally, there are hopfian BS groups which are not solvable. – Moishe Kohan Sep 26 '16 at 13:28