Is $\langle\mathbb R^*,*\rangle$ a group ($\mathbb R^* =\mathbb R\setminus\{0\}$), where $*$ is defined as $a*b = |a| b$?

Question

Let $\mathbb R^*$ be the set of all real numbers except $0$. Define $*$ on $\mathbb R^*$ by $a*b= | a | b$.

$*$ is associative on $\mathbb R^*$ and $1$, $-1$ are left identities and $1$ is right identity for $a>0$ and $-1$ is right identity for $a<0$.

The question is, will $\langle\mathbb R^*,*\rangle$ be a group? Apart from this, I also want to know, will the be a unique identity for this binary operation?

Answer 1

The only elements $a\in\mathbb R\setminus\{0\}$ such that$$(\forall b\in\mathbb R\setminus\{0\}):a*b=b$$are $\pm1$. But for none of them it is true that$$(\forall b\in\mathbb R\setminus\{0\}):b*a=b.$$Therefore $(\mathbb R\setminus\{0\},*)$ has no identity element and so it is not a group.