An example of a group such that $G \cong G \times G$

Question

I was trying to find an example such that $G \cong G \times G$, but I am not getting anywhere. Obviously no finite group satisfies it. What is such group?

Answer 1

Take $$G = H \times H \times H \times \cdots$$ for $H$ any nontrivial group.

Answer 2

Let $G = \mathbb Z ^ \mathbb N$ (with pointwise addition as the product). Then let $f:G \times G \longrightarrow G$ be $$f(g,h)(n) = \begin{cases} g(k), &n = 2k \\ h(k), &n = 2k+1 \end{cases}$$ You can verify $f$ is an isomorphism.

Answer 3

Let $G$ be the trivial group, for the only finite example.

Answer 4

I think it is an open problem whether or not there exists a finitely presented group $G$ satisfying $G \simeq G \times G$. However, several such finitely generated groups are known. Probably the first example was given by Jones in Direct products and the Hopf property.

Answer 5

As others have mentioned, the trivial group satisfies this property for reasons that are mostly unrelated to group theory. If a category has binary products and a terminal object $1$ then $A \times 1 \cong A$ in a canonical way. Of course, we also have $A \times B \cong B \times A$ canonically, so in fact a terminal object serves as a unit for the product. Thus, $1 \times 1 \cong 1$.

In the case of the category of groups we have products and a terminal object (the trivial group) so this holds.