# Group action on Grasmannian

A book i'm reading introduces the notion of a group $$G$$ acting on a set $$X$$ and then lists these two examples. I have two questions:

Example 1) The group $$GL(V)$$ of linear bijections from a vector space $$V$$ to itself

Example 2a) The group O($$E$$) of linear bijections that are also isometries from a Euclidean space $$E$$ to itself.

Example 2b) The book then defines $$G_{E,p}$$ to be the set of $$p$$-dimensional subspaces of $$E$$. And says that O($$E$$) also acts on $$G_{E,p}$$.

Question 1) Does the group $$GL(V)$$ act on the corresponding set $$G_{V,p}$$? Or do we somehow need $$V$$ to carry a Euclidean structure?

Question 2) The book then introduces the notion of a stabilizer of an element under a group action, and says that for $$X = G_{E,p}$$, we have that the stabilizer of an element of this set is isomorphic to $$O(p)xO(n-p)$$ where $$O(p) = O(\mathbf{R^p})$$.

Can someone clarify this?

For 1) Yes. An element $$\phi \in \mbox{GL}(V)$$ must send a basis of $$W\in G_{V,p}$$ into $$p$$ linearly independent vectors which generate $$\phi(W)$$.
Now for 2) fix $$W \in G_{E,p}$$. We have that $$E = W \oplus W^\perp$$ and any $$\phi \in O(E)$$ satisfying $$\phi(W) = W$$ must satisfy $$\phi(W^\perp) = W^\perp$$. Chosing an appropriate basis for $$E$$ we see that $$\phi = \begin{pmatrix} A & 0 \\ 0 & B \end{pmatrix}$$ where $$(A,B) \in \mbox{GL}(W)\times \mbox{GL}(W^\perp)$$. From $$\phi$$ being orthogonal, it follows that $$(A,B) \in O(W)\times O(W^\perp)\simeq O(p)\times O(n-p).$$