What does $g^1$ represent in group theory?

I was going through the definition of "order of an element" and in the examples following it the author had considered the group $U(15)$ under multiplication modulo $15$, where it was calculating the order of each element there it was given, $7^1 = 7$, $7^2 = 4 $... Now I understand that $7^1 = 7\pmod{15} = 7$. But In group theory where we talk about binary operations what does $g^1$ even mean? Don't we need two elements to perform Binary operation.

Generally if $G$ is a group with the neutral element $e\in G$, $g\in G$ is any element and $n\in\mathbb{Z}$ is an integer then $g^n$ is defined by the following rules:

$$g^0:=e$$ $$g^n:=gg^{n-1}\text{ for }n>0$$ $$g^n:=(g^{-1})^{-n}\text{ when }n<0$$

The middle rule is recursive. In the last one note that when $n<0$ then $-n>0$ and so $(g^{-1})^{-n}$ is well defined by the previous rule.

Few examples that follow from this definition:

$$g^1=gg^0=ge=g$$ $$g^2=gg^1=gg$$ $$g^3=gg^2=ggg$$ $$g^{-2}=(g^{-1})^2=g^{-1}g^{-1}$$ $$g^{-3}=(g^{-1})^3=g^{-1}g^{-1}g^{-1}$$

and so on.