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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.

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You are doing well for a and b. For c, do you know the orbit stablizer theorem? –  Sanchez Apr 19 '13 at 5:12
    
+1 for a well thought out question –  Michael Joyce Apr 19 '13 at 5:13
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1 Answer

up vote 2 down vote accepted

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 $B$?

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Thank you very much! It is all clear now! –  Orest Xherija Apr 19 '13 at 5:15
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