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
Famous quotes containing the words proof, group and/or theory:
“The chief contribution of Protestantism to human thought is its massive proof that God is a bore.”
—H.L. (Henry Lewis)
“Jury—A group of twelve men who, having lied to the judge about their hearing, health, and business engagements, have failed to fool him.”
—H.L. (Henry Lewis)
“Won’t this whole instinct matter bear revision?
Won’t almost any theory bear revision?
To err is human, not to, animal.”
—Robert Frost (1874–1963)