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!
##SO(2)## is the special orthogonal group in two dimensions, which is a mathematical group that represents rotations in a two-dimensional space. It relates to the isotropy group for ##x \in R^3## because it is the group of rotations that leave a fixed point (the origin) unchanged in a three-dimensional space.
In the context of ##x \in R^3##, isotropy refers to the property of a point or a region being invariant under a certain group of transformations. In this case, the isotropy group for ##x \in R^3## is the group of rotations (represented by ##SO(2)##) that leave ##x## unchanged.
If a point has isotropy under ##SO(2)##, it means that the point remains unchanged under any rotation in the two-dimensional plane. In other words, the point is invariant under the group of rotations represented by ##SO(2)##.
Understanding ##SO(2)## as the isotropy group for ##x \in R^3## has various applications in fields such as physics, engineering, and computer graphics. For example, it is used in robotics to control the orientation of objects, in computer graphics to create realistic 3D animations, and in physics to study the symmetries of physical systems.
##SO(2)## being the isotropy group for ##x \in R^3## means that any rotation in the two-dimensional plane will preserve the symmetry of ##x##. This is significant because it allows us to analyze the symmetries of objects in three-dimensional space by using the simpler two-dimensional rotations represented by ##SO(2)##.