Let G be a cyclic group of order n. Prove that g" = e for every element g in G. Use the fact that the order of any element in a cyclic group divides the order of the group. Let G be a group with more than one element. If the only subgroup of G are (e) and G, prove that G is cyclic and has prime order. For the first claim, pick an arbitrary element g of G and consider For the second claim, use the Fundamental Theorem of Cyclic Groups.