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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.