Suppose that m ∈ Z≥1, Cm is a cyclic group of order m, and φ∶ Cm → Cm is an isomorphism. Prove that if a generates Cm then φ(a) generates Cm.

Question

I am kind of confused on how to go about this? I know that there has to be some element a that if gives us Cm and we know its isomorphic so it must be for both but I don't know how to go about it.

Answer 1

Hint: Consider the subgroup $H$ of $C_m$ that $\phi(a)$ generates. What can you say about $\phi^{-1}(H)$?

Answer 2

Let $b\in C_m$; then, by assumption, $\varphi^{-1}(b)=a^k$, for some integer $k$.

Then $b=\dots$