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
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $6$: Cosets: Exercise $14$