Quotient groups of $\mathbb{RxR}$
In each of the following, H is a subset of $\mathbb{RxR}$.
a) Prove that H is a normal subgroup of $\mathbb{RxR}$ (Remember that
every subgroup of an abelian group is normal)b) In geometric terms, describe the elements of the quotient group
$G/H$c) In geometrical terms or otherwise describe the operation of $G/H$
where $H=\{(x,y):y=-x\}$
This is what I've done so far:
a) Since $\mathbb{RxR}$ is abelian and we are given that H is a subset, then we must show that H is a subgroup first. This means that H has an inverse and is closed with respected to addition.
H has an identity $(0,0)$ and an inverse $(-a,-b)$ for $a,b \in H$. Also, for $a,b,c,d \in H$. then we have: $(a,b)+(c,d)=(a+c,b+d)=(a+c,-a+-c)$ which is in $H$, therefore closed with respect to addition.
Thus, $H$ is a subgroup
The thereom that states every subgroup of an abelian group is normal. We know $H$ is normal since $\mathbb{RxR}$ is abelian
b) To describe the elements of the quotient group $G/H$, I started with the definition. $G/H$ is the set of cosets of $H$ in $G$. For this case, I am going to stick with right cosets. I am not sure what this is asking for, so I decided to list some elements.
We know that $G$ be in the form $(x,y),(-x,-y),(-x,y),(x,-y)$ where $x,y$ are integers in $G$ The reals also contain the rationals so we can also have $(\frac{a}{b},y)$ and so on …. where $a,b$ are integers as well
So the cosets $H+g$ where $g \in G$ will be the same. It can contain (0,0), positive or negative integers and positive or negative rationals.
So I'm not sure how I can describe this actually, if anyone can help with this part.
c) I believe the operation for $G/H$ is addition.
Best Answer
Hint: in particular, $H$ is an element (the zero element) of $G/H$. What does it look like geometrically? Can you graph it? Once you know that, the other cosets are just translates of $H$, so they should look the same, just moved around. For instance, what does the coset $(0,1) + H$ look like?