Proofs of Fermat's Little Theorem - Proof Using Group Theory

Proof Using Group Theory

This proof requires the most basic elements of group theory.

The idea is to recognise that the set G = {1, 2, …, p − 1}, with the operation of multiplication (taken modulo p), forms a group. The only group axiom that requires some effort to verify is that each element of G is invertible. Taking this on faith for the moment, let us assume that a is in the range 1 ≤ a ≤ p − 1, that is, a is an element of G. Let k be the order of a, so that k is the smallest positive integer such that

By Lagrange's theorem, k divides the order of G, which is p − 1, so p − 1 = km for some positive integer m. Then