is complex number under absolute value a group? I have just started going over abstract algebra.

One of the question is 

$*$ is defined on $\mathbb C$ such that $a*b=|ab|$

I tried to check three axioms : 1) Associativity 2) identity 3) inverse

I found out with long computation that Associativity works.

For identity, $a*e=e*a=a$ , where $a*e=|ae|=a$

I am quite confused how to find out the value of e, since this is a complex number.

Can anyone show me why this is a group or not a group?
It’s not a group as it fails to have an identity.

Hint. Look at $a = -1$ and the range of the defined operation “$*$”.
It is not group. Problem is with a=-s s$\in R^+$.