# Thread: Elements of quotient groups

1. ## Elements of quotient groups

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

2. ## Re: Elements of quotient groups

Originally Posted by MachinePL1993
Hello,
I have a group $G=(\mathbb{R}^{2},+)$ and it's subgroup $H=\{{(x,y) \in \mathbb{R}^{2}: x+3y=0\}$. What are the elements of quotient group $G / H$? How to find them?
Hi MachinePL1993!

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

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