##SO(2)## is the isotropy group for a vector ##x \in R^3## when considering the action of ##SO(3)## on ##R^3## through rotations. The isotropy group consists of all elements that fix a specific point or vector, which in this case is the axis of rotation. Therefore, while ##SO(2)## represents rotations in a plane, it specifically fixes the line connecting the origin to the point ##x## when ##x## lies along the axis of rotation. This understanding clarifies the relationship between ##SO(2)## and the isotropy group in three-dimensional space.
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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.Silviu said: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!