You got a vector $x \neq 0$ with $x \in \mathbb{R}^3$. I have to show that there is a bijection between the stabilizer $\{A \in O_3(\mathbb{R}) \mid Ax = x \}$ and $O_2(\mathbb{R})$. Can someone tell me what the stabilizer looks like and how he is related with the $O_2$ group ? (The group action is the standard matrixproduct as you see)
Bijection between $O_2(\mathbb{R})$ and the stabilizer of an $O_3(\mathbb{R})$ matrix
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group-theory
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3HINT: If you think of $\Bbb R^2$ as the $xy$-plane in $\Bbb R^3$, the stabilizer of $\langle 0,0,1\rangle$ is $O_2(\Bbb R)$. – 2012-09-30
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0thanks. This helps a lot :) – 2012-09-30