What Does $(U_{n},+)$ mean?

Question

I am having a problem understanding what $(U_{n},+)$ means. For clarification $U_{n}=\{[a]\in \mathbb{Z}_n:gcd(a,n)=1\}$ where $[a]$ denotes the equivalence class and $\mathbb{Z}_n$ is the set of elements$\{0,1,2,3,...,n-1\}$. My question is, Can we perform addition on a group of units. looking at the definition, the group of units for any integer does not contain $0$ and since $0$ is the identity element in addition There can't be a group formed from a Cayley table of $(U_{n},+)$. For example for $(U_{5})=\{1,2,3,4\}$ and performing addition on it will not make sense to me so what does it mean to say $(U_{n},+)$. I saw this from a worksheet that a professor provided to his students.

Answer 1

The group of units of the ring $\Bbb Z/n\Bbb Z$ is either denoted by $(\Bbb Z/n\Bbb Z)^{\times}$, or by $U(n)$ or by $U_n$. In each case, the group operation is multiplication. However, the ring has also an abelian group with addition, namely $(\Bbb Z/n\Bbb Z,+)$. So I suppose this was meant here. So the confusion is between $(\Bbb (Z/n\Bbb Z)^{\times},\cdot)$ and $(\Bbb Z/n\Bbb Z,+)$. A further possibly is, that the question was, whether or not $(U_n,+)$ is a group, and the answer is "no".

Answer 2

In all likelihood the group of units, or multiplicative group of integers modulo $n$ is being referred to. That's $\Bbb Z_n^×$.

Not to be confused with $U(n)$, the unitary group, consisting in $n×n$ unitary matrices.

Our group $U_n$ consists in the integers relatively prime to $n$, with group operation multiplication modulo $n$.

For instance, whenever $p$ is prime, we get $U_n=\{1,2,\dots, p-1\}$.

In general, $U_n$ has order $\varphi (n)$, where $\varphi $ is Euler's totient function.

You can check that the group axioms are indeed satisfied.