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
Read more about this topic: Proofs Of Fermat's Little Theorem
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