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

Theorem
Let $\left({G, *}\right)$ be an abelian group.

Let $\left({\Z, +, \times}\right)$ be the ring of integers.

Let $\circ$ be the mapping from $\Z \times G$ to $G$ defined as:


 * $\forall n \in \Z: \forall x \in G: n \circ x = *^n x$

where $*^n x$ is the $n$th power of $x$:


 * $*^n x = \underbrace{x * x * \ldots * x}_{\text{$n$ copies of $x$}}$

Then $\left({G, *, \circ}\right)_\Z$ is a unitary $\Z$-module.

This is called the $\Z$-module associated with $G$.

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

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

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

From the definition, $n \circ x = x^n$ and so $n \circ \left({x * y}\right) = \left({x * y}\right)^n$

From Power of Product in Abelian Group, $\left({x * y}\right)^n = x^n * y^n = \left({n \circ x}\right) * \left({n \circ y}\right)$.

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

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

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

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

That is, that $x^{nm} = \left({x^m}\right)^n$.

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

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

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

This follows from Power of Group Element.

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