Chapter 4.4, Problem 26E

Let $p$ be prime and $G$ the multiplicative group of units $U_p=\{[a]\in Z_p|[a]\ne [0]\}$. Use Lagrange’s Theorem in $G$ to prove Fermat’s Little Theorem in the form ${[a]}^p=[a]$ for any $a\in Z$. (compare with Exercise 54 in section 2.5)

Let $p$ be a prime integer. Prove Fermat’s Little Theorem: For any positive integer $a,a^p\equiv a(\mathrm{mod}p)$. (Hint: Use induction on $a$, with $p$ held fixed.)