Multiplication Tables for the Group of Integers Modulo n
Given a positive integer , the set of positive integers coprime to satisfies the axioms for an Abelian group under the operation of multiplication modulo . For instance, and because . This Demonstration shows the array plot of the multiplication table modulo corresponding to .
The order of is given by Euler's totient function , implemented in Mathematica as EulerPhi[n], which for has values . is cyclic only if is , or , where is an odd prime and . The first few values for which is not cyclic are . Any generator in the cyclic case is called a primitive root modulo .