Let $G$ be a finite abelian group, and let $n$ divide $|G|$. Let $m$ be the number of solutions of $x^n=1$. Prove that $n\mid m$.

By the fundamental theorem of finite abelian groups we may choose a $G$ subgroup $G_n$ of size $n$. Lagrange's theorem gaurantees $G_n\leq\ker(\varphi_n)$ where $\varphi_n$ denotes the $G$ endomorphism $\varphi_n:x\mapsto x^n$ and $\leq$ denotes subgroup inclusion. Finally, by Lagrange's theorem once again, $$n=|G_n|\;\Big\vert\;|\ker(\varphi_n)|=m$$