It is given that G is a group. Therefore the inverse must exist. Let us find the identity element first to find the inverse.
We know,
a*e=e*a=a (where e is the identity element)
we know a*e = a+e-ae
=> a+e-ae=a
=> e=0
since identity is 0 ,we can say that (by group property)
a*b=b*a=0 (i.e identity element)
a+e-ae=0
=>e=$\frac{a}{a-1}$ and since a!=0 , this statifies
so inverse of -2 should be
e=$\frac{-2}{-2-1}$ = $\frac{2}{3}$
Just to check put a=-2 and e=2/3 in a*b and you should be getting a 0.