Prove that any two cyclic groups of the same order are isomorphic.
Hint: Let G and G' be cyclic groups with generators a and a' respectively. If G and G' are both infinite show that defined by for is an isomorphism. If G and G' are both of finite order n, use again but note that it is necessary to prove that is consistently defined because it is possible to have when i does not equal j.
It's easy to show that phi is injective when the group is infinite since a^m = a^n means that m = n in a cyclic group of infinite order, but I am having trouble showing phi is injective with the finite case and any help would be appreciated. How do I show it is consistently defined? Thanks again everyone