Consider the rationals $\mathbb{Q}$ viewed as group under addition and let $G=\mathbb{Q}/\mathbb{Z}$

Question

Consider the rationals $\mathbb{Q}$ viewed as group under addition and let $G=\mathbb{Q}/\mathbb{Z}$ where $ \mathbb{Z} \;\triangleleft \; \mathbb{Q}$

A) What is order of $\frac{3}{5}$

B) Show that G has a cyclic subgroup of order n for each $n>1$

Please help to solve this problem i really stuck

HINT: a) Look for the least positive integer $n$ such that $n\left(\frac35\right)$ is an integer. — Ángel Mario Gallegos, Dec 06 '16 at 16:07
@DietrichBurde I think this question is sufficiently distinct to be considered a non-duplicate — Ben Grossmann, Dec 06 '16 at 16:07

Ben Grossmann · Answer 1 · 2016-12-06T16:07:54.673

5

Hint: The order of an element $q \in \Bbb Q /\Bbb Z$ is the lowest integer $n$ such that $$ \overbrace{q + q + \cdots + q}^{n \text{ times}} = 0 $$ It helps to think of $\Bbb Q /\Bbb Z$ as the rationals "modulo $1$".

edited Dec 06 '16 at 16:07

answered Dec 06 '16 at 16:03

Ben Grossmann

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I think you meant the rationals "modulo 1", not the integers – Alan Dec 06 '16 at 16:07
@Alan good catch! – Ben Grossmann Dec 06 '16 at 16:08

Consider the rationals $\mathbb{Q}$ viewed as group under addition and let $G=\mathbb{Q}/\mathbb{Z}$

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