The following question is Question 12.4 from Abstract Algebra by Dan Saracino.
Suppose $G=Z_2 \times Z_2 \times Z_2 \times Z_2 \times \cdots $, where $Z_2$ is the additive group of integers modulo 2.
How to prove or disprove that $G$ is isomorphic to $G \times G$?
This is my attempt:
I have tried to find a function which is an isomorphism.
If $\phi(x)=(x,x)$, then $$\phi(a+b)=(a+b,a+b)=(a,a)+(b,b)=\phi(a) \circ \phi(b)$$
Also, $a\ne b$ implies $(a,a) \ne(b,b)$ and so $\phi(a) \ne \phi(b)$.
Now, I do not know how to prove this function is onto.
I also do not know if we consider $x=(x_1,x_2, \cdots)$, whether $(x,x)$ is well defined or not.