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:

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

$\blacksquare$


Sources