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Problem 91 (The Order Of An Element Divides The Order Of A Group)

Let $G$ be a finite group. Use Lagrange's theorem to prove the following two corollaries.

  1. Let $a\in G$. The order of $a$ divides the order of $G$.
  2. If $|G|=p$ for some prime $p$, then $G$ is cyclic.

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