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:
“He who has never failed somewhere, that man can not be great. Failure is the true test of greatness. And if it be said, that continual success is a proof that a man wisely knows his powers,—it is only to be added, that, in that case, he knows them to be small.”
—Herman Melville (1819–1891)
“No group and no government can properly prescribe precisely what should constitute the body of knowledge with which true education is concerned.”
—Franklin D. Roosevelt (1882–1945)
“The human species, according to the best theory I can form of it, is composed of two distinct races, the men who borrow and the men who lend.”
—Charles Lamb (1775–1834)