multiples of n). Outside of number theory the simpler notation is often used, though it can be confused with the -adic integers when n is a prime number. The multiplicative group of integers modulo n, which is the group of units in this ring, may be written as (depending on the author) (for German Einheit, which translates as unit), , or similar notations. This article uses The notation refers to the cyclic group of order n. It is isomorphic to the group of integers modulo n under addition. Note that or may also refer to the group under addition. For example, the multiplicative group for a prime p is cyclic and hence isomorphic to the additive group , but the isomorphism is not obvious. Structure The order of the multiplicative group of integers modulo n is the number of integers in coprime to n. It is given by Euler's totient function: . For prime p, . Cyclic case The group is cyclic if and only if n is 1, 2, 4, pk or 2pk, where p is an odd prime and . For all other values of n the group is not cyclic. This was first proved by Gauss. This means that for these n: where By definition, the group is cyclic if and only if it has a generator g (a generating set {g} of size one), that is, the powers give all possible residues modulo n coprime to n (the first powers give each exactly once). A generator of is called a primitive root modulo n. If there is any generator, then there are of them. Powers of 2 Modulo 1 any two integers are congruent, i.e., there is only one congruence class, [0], coprime to 1. Therefore, is the trivial group with element. Because of its trivial nature, the case of congruences modulo 1 is generally ignored and some authors choose not to include the case of n = 1 in theorem statements. Modulo 2 there is only one coprime congruence class, [1], so is the trivial group. Modulo 4 there are two coprime congruence classes, [1] and [3], so the cyclic group with two elements. Modulo 8 there are four coprime congruence classes, [1], [3], [5] and [7]. The square of each of these is 1, so the Klein four-group. Modulo 16 there are eight coprime congruence classes [1], [3], [5], [7], [9], [11], [13] and