Multiplicative order - Definition and Overview

In number theory, given an integer a and a positive integer n with gcd(a,n) = 1, the multiplicative order of a modulo n is the smallest positive integer k with

a^k ≡ 1 (modulo n).

The order of a modulo n is usually written ord_n a, or O_n(a).

For example, to determine the multiplicative order of 4 modulo 7, we compute 4² = 16 ≡ 2 (modulo 7) and 4³ ≡ 4×2 = 8 ≡ 1 (modulo 7), so ord₇(4) = 3.

This notion is a special case of the order of group elements: if (G, *) is a group written with the usual multiplicative notation (so that a^t represents the t-fold product under *), the order of the element a of G is the least positive integer k such that a^k=e (where e denotes the identity element of G). The multiplicative order of a number a modulo n is then nothing but the order of a in the group U(n), whose elements are the residues modulo n of the numbers coprime to n, and whose group operation is multiplication modulo n. This is the group of units of the ring Z_n; it has φ(n) elements, φ being Euler's totient function.

For general reasons then, as a case of Lagrange's theorem, ord_na always divides φ(n). If ord_n a is actually equal to φ(n) and therefore as large as possible, then a is called a primitive root modulo n. This means that the group U(n) is cyclic and the residue class of a generates it.

Not every number n has a primitive root modulo n, but prime numbers always do. If a number n admits a primitive root modulo n, then there are φ(φ(n)) different residue classes modulo n which serve as primitive roots. This is an instance of a general statement about the number of generators of cyclic groups.

See also: Modular arithmetic, order (group theory)