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Consider the quotient group $\mathbb{Q}/\mathbb{Z}$ of the additive group of rational numbers. Then how to find the order of the element $2/3 + \mathbb{Z} $ in $\mathbb{Q}/\mathbb{Z}$.

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2 Answers 2

up vote 1 down vote accepted

Supplementing Makoto's answer, we can generalize: the order of an element $p/q \in \mathbb{Q}/\mathbb{Z}$ is the least integer $n$ such that $(p/q)n$ is an integer. If the fraction is written in reduced form, this is the same as the denominator.

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Let $a = 2/3 + \mathbb{Z}$. Since $3a = 0$ and $a \neq 0$, the order of $a$ is $3$.

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@ram: Note that in Makoto's nice answer(+1); $3a=0=0_{\mathbb Q/\mathbb Z}=\mathbb Z$. –  B. S. Sep 16 '12 at 9:26

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