I Understanding ##SO(2)## as Isotropy Group for ##x \in R^3##

[Silviu]

Hello! I am not sure I understand why $SO(2)$ is the isotropy group for $x \in R^3$. If I understood it well, the isotropy group contains all the elements such that $gx=x$. But this is not the case for $SO(2)$ as this group represents rotations in a plane, so unless x is the axis of rotation, x will be changed. What am I getting wrong here? Thank you!

 

[George Jones]

I am not sure of your context, but one interpretation is that $x$ represents the line (or vector) that connects the origin to $x$ considered as a point.

 

[WWGD]

Hello! I am not sure I understand why $SO(2)$ is the isotropy group for $x \in R^3$. If I understood it well, the isotropy group contains all the elements such that $gx=x$. But this is not the case for $SO(2)$ as this group represents rotations in a plane, so unless x is the axis of rotation, x will be changed. What am I getting wrong here? Thank you!

If I understand your question correctly, you must first have a group action defined, after which you determine the subgroup that fixes a given element.

 

[lavinia]

If you mean the action of $SO(3)$ on $R^3$ by rotations then any rotation fixes its axis of rotation. The subgroup of rotations with a given axis is isomorphic to $SO(2)$