Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.16
Use Lagrange’s Theorem in the multiplicative group $(\mathbb{Z}/(p))^\times$ to prove Fermat’s Little Theorem: if $p$ is prime then $a^p \equiv a \pmod p$.
Solution: If $p$ is prime, then $\varphi(p) = p-1$ (where $\varphi$ denotes the Euler totient). Thus $$|((\mathbb{Z}/(p))^\times| = p-1.$$ So for all $a \in (\mathbb{Z}/(p))^\times$, we have $|a|$ divides $p-1$. Hence $$a = 1 \cdot a = a^{p-1} a = a^p \pmod p.$$
