Is there a *finitely generated* nontrivial group $G$ such that $G \cong G \times G$?


Here are some properties which such a group $G$ has to satisfy:


* $G$ is not abelian (otherwise $G$ is a noetherian $\mathbb{Z}$-module, and the composition of the first projection $G \times G \to G$ with an isomorphism $G \cong G \times G$ will be bijective, i.e. $G$ is trivial).
* $G$ is perfect (apply the first observation to $G/G'$)



When is $G$ isomorphic to $G \times G$?
Yes. Some Googling turns up [J. M. Tyrer Jones, "Direct products and the Hopf property," *J. Austral. Math. Soc.* **17** (1974), 174-196](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=4894728).


