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.