The Ultimate Multiplicative group of integers modulo n Dog Breeds Information Guide and Reference

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 $\mathbb{Z}/n\mathbb{Z}$ for a given integer $n > 1$ . It is often denoted $\mathbb{Z}/n\mathbb{Z}^*$ .

The order of the group is given by Euler's totient function. Thus for $n$ prime, the order of the group is $n - 1$ .

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.

The multiplicative group is a cyclic group if and only if $n = 2$ , $n = 4$ , $n = p m$ , or $n = 2 p m$ 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).

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 $C_2 \oplus C_{2^{k-2}}$ .

Categories: Number theory | Group theory