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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?

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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 $$(A,B) \in O(W)\times O(W^\perp)\simeq O(p)\times O(n-p).$$

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