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 ?