TITLE: Multiplicative group of integers modulo n definition issues
QUESTION [8 upvotes]: It is easy to verify that the set $(\mathbb{Z}/n\mathbb{Z})^\times$ is closed under multiplication in the sense that $a, b ∈ (\mathbb{Z}/n\mathbb{Z})^\times$ implies $ab ∈ (\mathbb{Z}/n\mathbb{Z})^\times$, and is closed under inverses in the sense that $a ∈ (\mathbb{Z}/n\mathbb{Z})^\times$ implies $a^{-1} ∈ (\mathbb{Z}/n\mathbb{Z})^\times$.

The question is the following:
Firstly, are $a$ and $b$ referring to each equivalence class of integers modulo $n$?
Secondly, by $a^{-1}$, what is this referring to? If $a$ is the equivalence class, I cannot see (or I am not sure) how I can make inverse set.

REPLY [2 votes]: The notation $a \in (\mathbb{Z} / n \mathbb{Z})^\times$ means the variable $a$ is being used to denote an element of $(\mathbb{Z} / n \mathbb{Z})^\times$.
If you are representing elements of $(\mathbb{Z} / n \mathbb{Z})^\times$ as equivalence classes, then $a$, being an element of $(\mathbb{Z} / n \mathbb{Z})^\times$, has a representation as an equivalence class.
The structure $(\mathbb{Z} / n \mathbb{Z})^\times$ is an abelian group. It has a group operation, an inverse, and a neutral element. As we are writing this group with multiplicative notation, we adopt the default notation ${}^{-1}$ for the inverse operation of the group.
As an aside, I think focusing on an object "being" an equivalence class really obscures the simplicity of what's going on. Equivalence classes are frequently just a technical set-theoretic trick.
One rather good notation for elements of the ring $\mathbb{Z} / n \mathbb{Z}$ is that elements are represented by integers. Not just integers in the range $\{ 0, 1, \cdots, n-1\}$, but any integer. The same element has many notations; for example, '3' and '10' are two separate notations for the same element of $\mathbb{Z} / 7 \mathbb{Z}$.
It may help to add a decoration to help keep track of when you are using '3' to denote an element of $\mathbb{Z}$ and when you are using it to denote an element of $\mathbb{Z} / 7 \mathbb{Z}$: common choices are $\overline{3}$ and $[3]$, or sometimes $[3]_7$. 
If you are representing the elements of $\mathbb{Z} / n \mathbb{Z}$ as equivalence classes, then the element $[3]$ is notation for the element represented by the equivalence class of 3. Despite that fact, you usually shouldn't be thinking "$[3]$ is an equivalence class": you should be thinking "$[3]$ is the element of $\mathbb{Z} / n \mathbb{Z}$ that comes from the integer 3".