The statement I am trying to prove is the following.
Let $k$ a field and $V$ a $k$-vector space of finite dimension. Let $\mathscr{B}$ be the set of ordered $k$-bases of $V$. The natural action of $GL(V)$ on $V$ induces an action on $\mathscr{B}$.
(a) Is this action transitive
Answer: I think I have managed to show that it is (unless I am terribly wrong).
(b) Describe the stabilizer of each element of $\mathscr{B}$.
Answer: I am finding the stabiliser to be trivial, but I think I might be making some mistake which I cannot identify. Maybe I am correct though.
(c) Show that this action induces a bijection of sets between $GL(V)$ and $\mathscr{B}$. Did you make any choice to construct such a bijection?
I don't have anything for the last one. If anyone would suggest a bijection, I could check myself that it is actually one.
Thank you all very much in advance for your time. Any help will be tremendously appreciated.
You've got (a) and (b), and (c) is asking you to put the two together.
Any time you have a group $G$ acting on a set $X$, if you pick a point $x \in X$, you get a bijection between the $G$-orbit of $x$ (often written as $G \cdot x$) and the set of left cosets of the stabilizer of $x$ (typically written as $G / G_x$).
Pick any ordered basis $B = \{v_1, \dots, v_n\} \in \mathscr{B}$. What does (a) tell you about the $GL(V)$-orbit of $B$? What does (b) tell you about the stabilizer of $B$?