quotient group

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April 3rd 2011, 02:50 PM
hizocar

quotient group

Consider $(\mathbb Q,+)$ and $(\mathbb Q/\mathbb Z,+).$ Let $n\ge1$ and define $G_n=\left( \left[ \dfrac{a}{n} \right],a\in \mathbb{Z} \right).$ Prove that $G_n$ is a subgroup of $\mathbb Q/\mathbb Z,$ and that it has order $n.$

$\left[\dfrac an \right]$ is the class of $\dfrac an.$ ******
April 3rd 2011, 04:24 PM
Drexel28

Quote:

Originally Posted by hizocar

consider the following groups:
$(\mathbb{Q}, +) (\mathbb{Q}/ \mathbb{Z}, +)$

And the group:
$G_{n}=([\frac{a}{n}] / a \in \mathbb{Z})$

proved that Gn is subgroup of Q / Z, and the order of Gn is n.

This doesn't make much sense, can you rephrase this? Define $G_n$ again...wite it out better.
April 3rd 2011, 04:36 PM
hizocar

now?
April 3rd 2011, 06:01 PM
hizocar

*****read the first post again, I've edited.******
April 4th 2011, 12:32 PM
Swlabr

Quote:

Originally Posted by hizocar

Consider $(\mathbb Q,+)$ and $(\mathbb Q/\mathbb Z,+).$ Let $n\ge1$ and define $G_n=\left( \left[ \dfrac{a}{n} \right],a\in \mathbb{Z} \right).$ Prove that $G_n$ is a subgroup of $\mathbb Q/\mathbb Z,$ and that it has order $n.$

$\left[\dfrac an \right]$ is the class of $\dfrac an.$ ******

Can you `get' the subgroup bit? For the order bit, I'll give you a hint...

HINT: $\left[\frac{a}{n}\right] = \left[\frac{a+n}{n}\right]$

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