# Isomorphism and Cyclic Groups

essie52

## Homework Statement

I need to prove that any isomorphism between two cyclic groups maps every generator to a generator.

2. The attempt at a solution
Here what I have so far:

Let G be a cyclic group with x as a generator and let G' be isomorphic to G. There is some isomorphism phi: G --> G'. Since phi is surjective then for any y in G' there exists some x in G such that phi(x) = y. Since x generates G then every element in x must be in the form of x^k for some integer k. Phi therefore, is determined by its value on x. The formula phi(x^k) = y^k defines the isomorphism.

This is the point where I go, "what now?" Any help appreciated! E

PS We have not discussed kernel in this class.