find a group with the property : a)find a nontrivial  group $G$ such that $G$ is isomorphic to $G \times G $
what i'm sure is that $G$ must be infinite ! but have now idea how to get or construct such group 
i chose many $G$'s but all of the homomorphism was not injective 
b) an infinite group in which every element has finite order but for each positive integer n there is an element of order n 
the group  $G = (Z_1 \times Z_2 \times Z_3 \times Z_4 \times ...) $ satisfies the conditions except the one which says that every element have finite order . 
how can we use this group to reach the asked group ? 

For the second problem, you can use the subgroup of  $\mathbb{Z}_1\times \mathbb{Z}_2\times \cdots\times \mathbb{Z}_n \times \cdot$ consisting of all sequences $(a_1,a_2,a_3,\dots)$ such that  all but finitely many of the $a_i$ are $0$.
