I am having trouble with showing that the function in this problem is injective. I've been trying it for a while already. Surjectivity wasn't hard, and neither was proving that it was a homomorphism.
Let $G$ be a finite abelian group and let $n$ be a positive integer relatively prime to $|G|$.
- Show that the mapping $\varphi(x)=x^n$ is an automorphism.
- Show that every $x\in G$ has an $nth$ root, i.e., for every $x$ there exists some $y \in G$ such that $y^n=x$.