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

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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.


Module Axiom $\text M 1$: Distributivity over Module Addition

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:

$\paren {x * y}^n = x^n * y^n = \paren {n \circ x} * \paren {n \circ y}$

$\Box$


Module Axiom $\text M 2$: Distributivity over Scalar Addition

We need to show that:

$\paren {n + m} \circ x = \paren {n \circ x} * \paren {m \circ x}$


That is:

$x^{n + m} = x^n * x^m$

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

$\Box$


Module Axiom $\text M 3$: Associativity

We need to show that:

$\paren {n \times m} \circ x = n \circ \paren {m \circ x}$

That is:

$x^{n m} = \paren {x^m}^n$

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

$\Box$


Unitary Module Axiom $\text {UM} 4$: Unity of Scalar Ring

We need to show that:

$\forall x \in G: 1 \circ x = x$

That is:

$x^1 = x$

This follows from the definition of Power of Group Element.

$\Box$


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

$\blacksquare$


Sources

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