# When is $G$ isomorphic to $G \times G$?

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 $G/G'$)
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apart from the trivial group, I guess. – wood Oct 27 '10 at 14:50
$G$ must also not be residually finite (as a finitely generated residually finite group is Hopfian, i.e. has no isomorphic proper quotients). – Jonathan Kiehlmann Nov 1 '10 at 17:59