# Show that the $GL(n,\mathbb R)/P_k$ is isomorphic to the $GL(n,\mathbb R)$-set grassmannian.

Let $$r be two positive integers and $$G=GL(n,\mathbb{R}).$$ If $$Gr(k,\mathbb{R}^n)$$ is the set of all $$k$$-subspaces, then show that the $$G$$-sets $$Gr(k,\mathbb{R}^n)$$ and $$G/P_k$$ is isomorphic, given $$P_k$$ is the subgroup formed by all blockwise triangular matrices in the form $$\begin{array}{l}\quad\left(\begin{array}{ll}A & B \\ 0 & D\end{array}\right) \\ \text { where } A \in G L(k, \mathbb{R}), D \in G L(n-k, \mathbb{R}), \text { and } B \in M_{k, n-k}(\mathbb{R})\end{array}$$.

Should I make a homomorphism such that $$P_k$$ is the kernel of it, then use the 1st isomorphism theorem to prove it? Or should I find an isomorphism from these two $$G$$-sets directly?

Note that $$P_k$$ is not normal and the Grassmannian is not a group. Instead, argue that $$GL(n)$$ acts transitively on $$Gr(k)$$ with stabilizer of one point $$X\in Gr(k)$$ (which one?) equal to $$P_k$$. Then construct a continuous bijection $$GL(n)/P_k\to Gr(k)$$, $$g\mapsto gX$$, $$g\in GL(n)$$. Then argue that the inverse is also continuous.
• @Tifsir: Not at all, the stabilizer o $\{0\}$ is the entire $GL(n)$. Try to work out the case $n=2$ first. – Moishe Kohan Sep 13 at 10:20