## Introduction:

Let $$\mathbb{Z}_n$$ be the subset of integers which are relatively prime to $$n$$. By definition, its cardinality is given by the Euler totient function:

\begin{equation} |\mathbb{Z}_n| = \phi(n) \end{equation}

and it may be easily shown that $$\mathbb{Z}_n$$ forms an Abelian group under multiplication modulo $$n$$.

## The Abelian group axioms:

The Abelian group axioms applied to $$(A,\cdot)$$ are as follows:

$$1.$$ Closure:

\begin{equation} \forall a,b \in A, a \cdot b \in A \end{equation}

$$2.$$ Associativity:

\begin{equation} \forall a,b,c \in A, (a \cdot b) \cdot c = a (b \cdot c) \end{equation}

$$3.$$ Identity element:

\begin{equation} \exists e \in A \forall a \in A, a \cdot e = e \cdot a = a \end{equation}

$$4.$$ Inverse element:

\begin{equation} \forall a \in A \exists b \in A, a \cdot b = b \cdot a = e \end{equation}

$$5.$$ Commutativity:

\begin{equation} \forall a,b \in A, a \cdot b = b \cdot a \end{equation}

## Proof:

We may demonstrate that $$(\mathbb{Z}_n, \cdot)$$ forms an Abelian group where $$\cdot$$ denotes multiplication modulo $$n$$.

$$1.$$ Multiplication modulo $$n$$, which results in congruence classes modulo $$n$$, is well-defined:

\begin{equation} a \equiv 1 \pmod{n} \iff \text{gcd}(a,n) = 1 \end{equation}

\begin{equation} a \sim b \iff a \equiv b \pmod{n} \end{equation}

$$2.$$ $$(\mathbb{Z}_n, \cdot)$$ is closed under multiplication modulo $$n$$:

\begin{equation} \text{gcd}(a,n) = 1 \land \text{gcd}(b,n) = 1 \implies \text{gcd}(ab,n) = 1 \end{equation}

$$3.$$ Integer multiplication respects the congruence classes:

\begin{equation} a \equiv a’ \pmod{n} \land b \equiv b’ \pmod{n} \implies ab \equiv a’b’ \pmod{n} \end{equation}

which implies that multiplication is associative, commutative, and the class of $$1 \in \mathbb{Z}_n$$ is the unique multiplicative identity.

$$4.$$ For any $$a \in \mathbb{Z}_n$$ the multiplicative inverse $$b \in \mathbb{Z}_n$$ exists since:

\begin{equation} \text{gcd}(a,n) = 1 \land \text{gcd}(b,n) = 1 \implies ab \equiv 1 \pmod{n} \end{equation}

and this completes the proof that $$(\mathbb{Z}_n, \cdot)$$ is Abelian.