The question states, Let G be the group of nonzero real numbers and let * be defined on G as a*b=a/b. Is G a group under *?
Would it then suffice to say that G is not a group because there is no unique identity such that a*e=e*a=a? Is there some other form of proof I should utilize? Thanks!
On
Yes, that's right. If we take an element from $G$, say $3$, then in order for identity to exist, we need to have $$\frac{e}{3} = \frac{3}{e} = 3$$ which has no solution because $\frac{e}{3} = 3 \implies e = 9$ however, $\frac{3}{e} = 3 \implies e = 1$, meaning that this equation system has no solution. Therefore $G$ has no identity element, so $G$ is not a group.
Yes, that works. You could also say $a/(b/c)$ vs $(a/b)/c$ for the lack of associativity.