The discussion revolves around the understanding of the special orthogonal group ##SO(2)## as the isotropy group for a point ##x## in ##R^3##. Participants explore the implications of group actions, particularly in the context of rotations and fixed points.
Participants express differing interpretations of the isotropy group and its relation to rotations, indicating that the discussion remains unresolved with multiple competing views.
There is a lack of clarity regarding the definitions and assumptions related to the group action and the nature of the point ##x## in ##R^3##, which may affect the understanding of the isotropy group.
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!