Let $(\Bbb R,*)$ be such that $x*y = |x|y$ for all $x,y \in \Bbb R$. Show that $(\Bbb R,*)$ is not a group. Let $(\Bbb R,*)$ be the mathematical system defined as: $x*y = |x|y$ for all $x,y \in \Bbb R$. Show that $(\Bbb R,*)$ is not a group.
Attempt:
I have shown the assosiative property.
Now, I have to find an identity element in $(\Bbb R,*)$.
Let $m \in \Bbb R$ be the identity element in $(\Bbb R,*)$. Let $y \in \Bbb R$. Then,
\begin{align*}
m*y &= y \\
|m|y &= y \\
m=1 &\vee m=-1
\end{align*}
and
\begin{align*}
y*m &= y \\
|y|m &= y \ldots (k)
\end{align*}
If $y \ge 0$, then $(k)$ becomes
$ym=y$ i.e. $m=1$.
If $y < 0$, then $(k)$ becomes
$-ym=y$ i.e. $m=-1$.
Thus, in two cases, we see that $m$ have two values. But, the identity element of a group must be unique. Hence, $(\Bbb R,*)$ has no an identity element i.e. is not a group. In particularly, $(\Bbb R,*)$ is a semigroup.
Is the above true?

Note that each group $G$ has exactly one idempotent, for if $x\in G$ such that $x^2=x$, then $x=x^{-1}x^2=x^{-1}x=e$.
But in $(\Bbb R,\ast)$, we have $(-1)\ast(-1)=|-1|(-1)=-1$ and $1\ast 1=|1|(1)=1$.
Hence $(\Bbb R,\ast)$ is not a group.
