Suppose that $\mathbb{G}$ is finite group of order $p$ ($p$ can be a prime number if necessary) and that $g$ is a generator of $\mathbb{G}$. What is the field of discrete logarithms of $g$?
I can see that the set $g^\mathbb{Z}\simeq \{1,\ldots p-1\}$ is finite and contains $p$ elements, it also has a commutative group structure with $0$ as neutral element, so it is (isomorphic to) $(\mathbb{Z}_{p-1},+)$. Addition is determined by $c:=a+b$ with $c$ determined by $g^c:= g^ag^b$. This is the discrete logarithm problem
But how can we deduces that there is also a multiplication? Can we derive it from the group law and the generator?
Because $(g^a)^b \neq g^{a \cdot_{\mathbb{Z}_p}b}$ in general.