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Problem 99 (Fermat's Little Lemma)

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

  1. We have $a^{|G|}=e$ for every $a\in G$.
  2. [Fermat's Little Lemma] Let $p$ be a prime and suppose $a\in U(p)$. Then we must have $a^p\pmod p=a$. (Hint: What is the order of $U(p)$?)

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