Group action : $ g * g^{-1} = e$?

Question

Okay so from group definition we know that if $(G,\times)$ is a grooup then : $g\times g^{-1} = e$. My problem is does the same hold for group action "$*$" ?
If $"*": G \times X \rightarrow X $ is an action , how can I prove :
$g*g^{-1} = e$?
My book does not change notation when referring to action and it's highly confusing to me because I feel like we use group operation properties as they hold for actions.
I tried to prove it myself but I am stuck:
By the first property of action: $e*g = g , \forall g \in G$
$\implies e*(g\times g^{-1}) = g \times g^{-1}$ and that's not helpful...

Answer 1

In a group action, $e * x =x$ and $(gh)*x = g*(h*x)$.

''how can I prove : $g∗g^{−1}=e$?''

The group can act on itself by left multiplication (here $X=G$): $g*h = gh$. Then $g*g^{-1} = gg^{-1}=e$ by definition of the inverse.