Isomorphism and Cyclic Groups: Proving Generator Mapping

[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.

 

[micromass]

So you got two cyclic groups G and G' and an isomorphism f:G-->G'.
Let x be a generator of G, then every g in G can be written as x^k=g.
You need to show that f(x) is a generator of G'. So pick an arbitrary h in G'. You'll need to find a k such that f(x)^k=h...