Question

Let $\mathit{G}$ be the additive group $\mathrm{ℝ}\times \mathrm{ℝ}$.

1. Show that $\mathbf{N}\mathbf{=}\mathbf{\left\{}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{|}\mathbf{y}\mathbf{=}\mathbf{-}\mathbf{x}\mathbf{\right\}}$ is a subgroup of $\mathbf{G}$.
2. Describe the quotient group $\mathit{G}\mathbf{/}\mathit{N}$ .

Short answer

It is proved that $N$ is a subgroup of $G$.

Step-by-step solution

Consider the elements

Consider two arbitrary elements as $(a,b),(c,d)\in N$. Apply the given definition of $N$, that is, $N=\{(x,y)\in G:y=-x\}$, we have role="math" localid="1654525463759" $b=-a$ and $d=-c$.

Prove the result

Now, assume the element as:

$\begin{array}{c}(a,b)+{(c,d)}^{-1}=(a,b)+(-c,-d)\\ =(a-c,b-d)\\ \in G\end{array}$

Then, we have:

$\begin{array}{c}b-d=-a+c\\ =-(a-c)\end{array}$

This implies that $(a,b)+{(c,d)}^{-1}\in N$.

Hence, $N$ is a subgroup of $G$.