We are learning about Quotient spaces, and group actions. For any $n\in\mathbb{N}$. We know that the function $G(n)=\{A\in\mathbb{R}^{n\times n}:A^TA=1\}$ is a group action on $\mathbb{R}^n$. I am struggling to see what the actual orbits are, and why $\mathbb{R}^n\backslash G(n)$ is homeomorphic to the positive real line. These are given as fact, but I dont see why these are elementary results.
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$\begingroup$ Have you tried this for $n=2$? $\endgroup$– Andres MejiaNov 6, 2018 at 1:59
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$\begingroup$ Your second sentence has some sloppy mathematical grammar. $G(n)$ is a collection of matrices, not a function. It describes (the underlying set of) a group, but you haven't described how it acts on $\mathbb{R}^n$. $\endgroup$– Qiaochu YuanNov 6, 2018 at 2:11
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$\begingroup$ You might want to check out math.stackexchange.com/questions/253179/… $\endgroup$– irchansNov 6, 2018 at 2:16
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$\begingroup$ the action is always "multiplication", as in if $G$ acts on $X$, then the action is the map $x\to gx$? Is this wrong? $\endgroup$– user593295Nov 6, 2018 at 2:17
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$\begingroup$ @user593295 I suggest that you read the wikipedia page on group actions en.wikipedia.org/wiki/Group_action . There are a lot of nice examples there. $\endgroup$– irchansNov 6, 2018 at 2:19
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