Disclaimer: I am new to the subject of Isomorphisms of Groups, so I may need some guidance here.
This is what I am working on:
Show that if $k$ and $n$ are relatively prime positive integers, then the function $f(x) = kx$ is an automorphism of $(\mathbb{Z_n},+)$.
I know that an isomorphism of a group with itself is called an automorphism. My textbook does not provide any examples of groups that are automorphic and that is the problem here. So for this problem, what are the conditions that I am suppose to know in order to carry out this problem? Any insights are greatly appreciated.
Hint: Prove that $f$ is an injective homomorphism.
(Make sure you understand why this suffices.)