# quotient group

• Apr 3rd 2011, 02:50 PM
hizocar
quotient group
Consider $\displaystyle (\mathbb Q,+)$ and $\displaystyle (\mathbb Q/\mathbb Z,+).$ Let $\displaystyle n\ge1$ and define $\displaystyle G_n=\left( \left[ \dfrac{a}{n} \right],a\in \mathbb{Z} \right).$ Prove that $\displaystyle G_n$ is a subgroup of $\displaystyle \mathbb Q/\mathbb Z,$ and that it has order $\displaystyle n.$

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

Originally Posted by hizocar
consider the following groups:
$\displaystyle (\mathbb{Q}, +) (\mathbb{Q}/ \mathbb{Z}, +)$

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

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 $\displaystyle G_n$ again...wite it out better.
• Apr 3rd 2011, 04:36 PM
hizocar
now?
• Apr 3rd 2011, 06:01 PM
hizocar
*****read the first post again, I've edited.******
• Apr 4th 2011, 12:32 PM
Swlabr
Quote:

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

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

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

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