AKA **unit group of $\mathbb{Z}/n\mathbb{Z}$** or **multiplicative group of integers modulo $n.$** > [!NOTE] Definition 1 ($(\mathbb{Z}/n\mathbb{Z})^{\times}$) > The set of [[Unit in Integers Modulo n|units]] of $\mathbb{Z}/n\mathbb{Z}$ ([[Integers modulo n|the integers modulo n]]). > [!NOTE] Definition 2 () > By [[Unit modulo n iff coprime to n|unit modulo n iff coprime]], $(\mathbb{Z}/n\mathbb{Z})^{\times}=\{ [a]_{m} \in \mathbb{Z}/n\mathbb{Z} \mid \gcd(a,n) =1 \}.$ **Equivalence**: # Properties **Algebra**: (1) By [[Unit Group of Ring is a Group|unit group is a group]], $((\mathbb{Z}/n\mathbb{Z})^{\times},\times_{n})$ is a group. (2) The [[Euler Totient Function|Euler totient function]] is defined as order of $(\mathbb{Z}/n\mathbb{Z})^{\times}$, denoted $\varphi(n).$ (3) In the case that the unit group is cyclic, its generators are known as [[Primitive Root Modulo n|primitive roots modulo n]]. Note that this is the case iff $n=1,2,4,p^{k}$ or $2p^{k}$ where $p$ is an odd prime and $k>0.$ # Applications **Cryptography**: [[RSA]].