Hello,

I have a group $\displaystyle G=(\mathbb{R}^{2},+)$ and it's subgroup $\displaystyle H=\{{(x,y) \in \mathbb{R}^{2}: x+3y=0\}$. What are the elements of quotient group $\displaystyle G / H$? How to find them?

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- Mar 17th 2013, 09:15 AMMachinePL1993Elements of quotient groups
Hello,

I have a group $\displaystyle G=(\mathbb{R}^{2},+)$ and it's subgroup $\displaystyle H=\{{(x,y) \in \mathbb{R}^{2}: x+3y=0\}$. What are the elements of quotient group $\displaystyle G / H$? How to find them? - Mar 17th 2013, 03:03 PMILikeSerenaRe: Elements of quotient groups
Hi MachinePL1993!

The elements of a quotient group are cosets, which are of the form (a + H) for an additive group.

In your case $\displaystyle a$ is point in $\displaystyle \mathbb{R}^{2}$ and H is your line.

When we add a point to a line, we get a parallel line.

So G/H consists of all lines parallel to H.