From Silverman and Tate Rational Points on Elliptic Curves
It is given that 3 does not divide $p-1$, the order of the cylic group $\mathbb{F}_p^*$ . It follows that the map $x \rightarrow x^3$ is an isomorphism from $\mathbb{F}_p^*$ to itself. . . .
If 3 divides the order of the group $\mathbb{F}_p^*$ , the map $x \rightarrow x^3$ is a homomorphism to itself that is neither one-to-one or onto.
Why does this follow? I know that $\mathbb{F}_p^*$ is also a field, but I don't know if that explains why this is (or isn't, respectively) a isomorphism since they seem to be treating $\mathbb{F}_p^*$ as a group, not a field.