Rational Numbers with Denominators Coprime to Prime under Addition form Group

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Theorem

Let $p$ be a prime number.

Let $\Q_p$ denote the set:

$\set {\dfrac r s : s \perp p}$

where $s \perp p$ denotes that $s$ is coprime to $p$.


Then $\struct {\Q_p, +}$ is a group.


Proof


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Taking each of the group axioms in turn:


Group Axiom $\text G 0$: Closure


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$\Box$


Group Axiom $\text G 1$: Associativity

As $\Q_p \subseteq \Q$ the result follows directly from Rational Addition is Associative and Restriction of Associative Operation is Associative.

$\Box$


Group Axiom $\text G 2$: Existence of Identity Element

By Integer is Coprime to 1:

$\dfrac 0 1 \in \Q_p$

regardless of our choice of $p$.


By the definition of addition on $\Q$:

$\dfrac a b + \dfrac 0 1 = \dfrac a b$

and

$\dfrac 0 1 + \dfrac a b = \dfrac a b$

for all $\dfrac a b \in \Q$.


Hence $\dfrac 0 1$ is the identity.

$\Box$


Group Axiom $\text G 3$: Existence of Inverse Element

For $\dfrac a b$ we have that $\dfrac {-a} b$ is the inverse of $\dfrac a b$.

As it has the same denominator as $\dfrac a b$ we have that $\dfrac {-a} b \in \Q_p$ as well.

$\blacksquare$


Sources

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