Z-Module Associated with Abelian Group is Unitary Z-Module

Theorem
Let $\struct {G, *}$ be an abelian group with identity $e$.

Let $\struct {G, *, \circ}_\Z$ be the $Z$-module associated with $G$.

Then $\struct {G, *, \circ}_\Z$ is a unitary $Z$-module.

Proof
The notation $*^n x$ can be written as $x^n$.

Let us verify that $\struct {G, *, \circ}_\Z$ is a unitary $\Z$-module by verifying the axioms in turn.

Axiom $(\text M 1)$
We need to show that $n \circ \paren {x * y} = \paren {n \circ x} * \paren {n \circ y}$.

From the definition, $n \circ x = x^n$ and so $n \circ \paren {x * y} = \paren {x * y}^n$

From Power of Product in Abelian Group:
 * $\paren {x * y}^n = x^n * y^n = \paren {n \circ x} * \paren {n \circ y}$

Axiom $(\text M 2)$
We need to show that $\paren {n + m} \circ x = \paren {n \circ x} * \paren {m \circ x}$.

That is, that $x^{n + m} = x^n * x^m$.

This is an instance of Powers of Group Elements: Sum of Indices.

Axiom $(\text M 3)$
We need to show that $\paren {n \times m} \circ x = n \circ \paren {m \circ x}$.

That is, that $x^{n m} = \paren {x^m}^n$.

This follows directly from Powers of Group Elements: Product of Indices.

Axiom $(\text M 4)$
We need to show that $\forall x \in G: 1 \circ x = x$.

That is, that $x^1 = x$.

This follows from the definition of Power of Group Element.

Having verified all four axioms, we have shown that $\struct {G, *, \circ}_\Z$ is a unitary $\Z$-module.