1. The problem statement, all variables and given/known data Title 2. Relevant equations 3. The attempt at a solution This would seem to be very easy problem, since it's intuitively obvious that if two groups are isomorphic, and one is cyclic, then the other is cyclic too. However, I can't seem to formalize it with math. Here is an idea. We can define that a group is cyclic by saying that for all b in G, there exists an a such that ##a^n = b## for some integer n. Now if G is cyclic, and we have an isomorphism ##\phi## from G to G', then it is true that ##\phi (a) ^n = \phi (b) = b'##, which means that G' is also cyclic. Does this sketch of a proof the right idea?