Condition for Element of Quotient Group of Additive Group of Reals by Integers to be of Finite Order

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Theorem

Let $\struct {\R, +}$ be the additive group of real numbers.

Let $\struct {\Z, +}$ be the additive group of integers.

Let $\R / \Z$ denote the quotient group of $\struct {\R, +}$ by $\struct {\Z, +}$.

Let $x + \Z$ denote the coset of $\Z$ by $x \in \R$.


Then $x + \Z$ is of finite order if and only if $x$ is rational.


Proof

From Additive Group of Integers is Normal Subgroup of Reals, we have that $\struct {\Z, +}$ is a normal subgroup of $\struct {\R, +}$.

Hence $\R / \Z$ is indeed a quotient group.


By definition of rational number, what is to be proved is:

$x + \Z$ is of finite order if and only if:
$x = \dfrac m n$

for some $m \in \Z, n \in \Z_{> 0}$.


Let $x + \Z$ be of finite order in $\R / \Z$.

Then:

\(\ds \exists n \in \Z_{\ge 0}: \, \) \(\ds \paren {x + \Z}^n\) \(=\) \(\ds \Z\) Definition of Quotient Group: Group Axiom $\text G 2$: Existence of Identity Element
\(\ds \leadstoandfrom \ \ \) \(\ds n x\) \(\in\) \(\ds \Z\) Condition for Power of Element of Quotient Group to be Identity
\(\ds \leadstoandfrom \ \ \) \(\ds n x\) \(=\) \(\ds m\) for some $m \in \Z$
\(\ds \leadstoandfrom \ \ \) \(\ds x\) \(=\) \(\ds \dfrac m n\)

$\blacksquare$


Sources

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