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

## 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
$x = \dfrac m n$

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 $\text G 2$: Existence of Identity Element $\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 \ \$ $\displaystyle x$ $=$ $\displaystyle \dfrac m n$

$\blacksquare$