Why is a "group action" defined as it is? Namely, a group $G$ acting on a set $S$, that is associative and has a unit element. This defines a monoid. But, authors go on to claim that the actions of $G$ are homomorphisms of $g\colon G \to \mathrm{Perm}(S)$. Permutations have inverses $g^{-1}$, which makes them into a group. So, why not state that an inverse exists in the definition?
In addition, consider a function acting on a vector space, say $\mathrm{GL}(n,V)$, which is the set of all "invertible" $n$-dimensional linear transformations on $V$.
Because it follows from the definitions.
Suppose that $G$ acts on a set $S$. Let $g \in G$. Then the map $S \rightarrow S$, $x \mapsto g \cdot x$ is a bijection, since it has a inverse: the map $x \mapsto g^{-1} \cdot x$.
To prove this, you need precisely the properties given by the definition of a group action: $1_G \cdot x = x$ and $(g_1 \cdot g_2) \cdot x = (g_1g_2) \cdot x$.