In mathematics, the multiplicative group of integers modulo n is the group defined by multiplication of the units (that is, the numbers relatively prime to n) in the ring for a given integer n > 1. It is often denoted . Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... The term group can refer to several concepts: In music, a group is another term for band or other musical ensemble. ... In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of... In mathematics, the integers a and b are said to be coprime or relatively prime if and only if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 1. ... In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ...

The order of the group is given by Euler's totient function. Thus for n prime, the order of the group is n − 1. In number theory, the totient φ(n) of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n. ...

This group has many applications in number theory and cryptography. In particular, by finding the size of the group, one can determine if n is prime: n is prime if and only if the size of the group is n − 1. See primality test. Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ... Cryptography portal Cryptography (from Greek kryptós, hidden, and gráphein, to write) is, traditionally, the study of means of converting information from its normal, comprehensible form into an incomprehensible format, rendering it unreadable without secret knowledge — the art of encryption. ... A primality test is an algorithm for determining whether an input number is prime. ...

The multiplicative group is a cyclic group if and only if n = 2, n = 4, n = p^m, or n = 2p^m for some odd prime p and some m > 0. For all other cases, the 2-torsion subgroup is not cyclic (i.e. has a quotient that is a Klein four-group). In mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that every element of the group is a power of a. ... In group theory, the torsion subgroup of an abelian group A is the subgroup of A consisting of all elements that have finite order. ... In mathematics, the Klein four-group (or just Klein group or Viergruppe, often symbolized by the letter V), named after Felix Klein, is a group with four elements, the smallest non-cyclic group. ...

Using the Chinese remainder theorem, once we determine the structure of the group for prime powers, we can determine the structure of the group for all n. By the above, the group is cyclic for odd prime powers. For n = 2^k, the structure of the group is . The Chinese remainder theorem is any of a number of related results in abstract algebra and number theory. ...