I am trying to prove the following using Lagrange's Theorem:
Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. Prove that if $x \in G$ and $\text{gcd}(|x|,|G:N|)=1$, then $x \in N$.
I know that $|N|$ divides $|G|$ and that $|x|$ divides $|G|$, but I am getting stuck trying to put these facts together. I am also having trouble understanding the significance of the gcd of $|x|$ and $|G:N|$ being 1.
Consider the canonical homomorphism $\pi\colon G\rightarrow G/N$, see the comment. Since $x^n=e$ implies $\pi(x)^n=[e]$ for a homomorphism $\pi$, the order of $\pi(x)$ divides the order of $x$, which is coprime to $|G/N|$ by assumption. Hence $|\pi(x)|=1$, which means $\pi(x)=[e]$, so that $x\in N$.