My professor wrote at the beginning of speaking about group actions this:
In general, Aut$(X) \subset $ Sym$(X)$ acts on $X$. If $G \subset Aut(X)$ is a subgroup, we say that "G acts on $X$ by appropriate automorphism."
Then he gave us a first definition for Group action which is: If $G$ a group, $X$ a set. a group action by $G$ on $X$ is a function : $G \times X \rightarrow X$ defined by $$(g,x) \mapsto {}^gx$$ such that $$ {}^g({}^hx) = {}^{(gh)}x$$ for all $g, h \in G.$
Then he gave us a second definition which is: any group homomorphism $G \rightarrow Aut(X).$
Then he gave us examples as follows:
If $V$ is a vector space over $k$ of dim. $n < \infty.$
1- $GL_{n}(k) = GL(V)$ acts on $V$ by linear transformations. $SL_{n}(k)$ acts on $V$ by restriction.
My questions are:
1- I do not understand how $Aut(X)$ acts on $X.$ what is the implied operation in that case?
2- I do not understand how is the second definition is also a group action definition? what is the implied operation in that case?
3- How can I prove that the example given is really a group action?
Could anyone help me answer those questions please?