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 insatiable thirst for everything which lies beyond, and which life reveals, is the most living proof of our immortality.”
—Charles Baudelaire (1821–1867)
“The boys think they can all be athletes, and the girls think they can all be singers. That’s the way to fame and success. ...as a group blacks must give up their illusions.”
—Kristin Hunter (b. 1931)
“The theory of rights enables us to rise and overthrow obstacles, but not to found a strong and lasting accord between all the elements which compose the nation.”
—Giuseppe Mazzini (1805–1872)