Hint
1) What is $\begin{bmatrix} x & x \\ x & x \end{bmatrix}\begin{bmatrix} y & y \\ y & y \end{bmatrix}$?
2) The multiplication of matrices is associative.
3) When you are looking for the identity you want
$$\begin{bmatrix} x & x \\ x & x \end{bmatrix}\begin{bmatrix} e & e \\ e & e \end{bmatrix}=\begin{bmatrix} x & x \\ x & x \end{bmatrix}$$
Now, do the multiplication on the left, what do you get?
4) With the $e$ from $3)$ solve
$$\begin{bmatrix} x & x \\ x & x \end{bmatrix}\begin{bmatrix} y & y \\ y & y \end{bmatrix}=\begin{bmatrix} e & e \\ e & e \end{bmatrix}$$
for $y$. Again, all you need to do is doing the multiplication...
P.S. In order for this to be a group, you need $x \neq 0$.
P.P.S Since $\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}=2\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$, you can prove that
$$F: \mathbb R \backslash\{0 \} \to G$$
$$F(x) =\frac{x}{2} \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$$
is a bijection and it preserves multiplications. Since $\mathbb R \backslash\{0 \}$ is a group it follows that G must also be a group and $F$ is an isomorphism... But this is probably beyond what you covered so far...
Closure:
Is it true that in the set $\{0\}$, for any two elements $a,b\in\{0\}$, the product $a\times b$ is also an element of $\{0\}$?
Answer:
Yes! Proof:
- If $a\in\{0\}$, then $a=0$
- If $b\in\{0\}$, then $b=0$.
- Therefore, $a\cdot b=0\cdot 0=0$.
- Therefore, because $0\in\{0\}$, we conclude $a\cdot b\in\{0\}$.
For associativity, a very similar argument can be made.
Identity:
Is $0$ the identity of $\{0\}$? That is, is it true that for any element $a\in\{0\}$, the element $a\cdot 0=a$?
Answer:
Yes! Proof:
- If $a\in\{0\}$, then $a=0$.
- Therefore, $a\cdot 0 = 0\cdot 0=0$.
- Since $a=0$ and $a\cdot 0=0$, we conclude $a\cdot 0=a$.
Best Answer
Since $N\lhd G$, $gN=Ng$ for all $g\in G$
Let $ab,hk\in HN$. Here note that $a,h\in H$ and $b,k\in N$.
Then $(ab)(hk)=a(bh)k$
Note that $Nh=hN$, so $bh=hc$ for some $c\in N$.
Thus $a(bh)k=a(hc)k=(ah)(ck)$ where $ah\in H$ and $ck\in N$.