 Solution

For a vector $x$, its isotropy group is the stabilizer subgroupFor a subgroup $H\le G$, its fixed-point subspace of a group action isIt is a linear subspace, the intersection of the kernels of $R_G(h)-I$. The subgroup fixes a particular vector pointwise; the fixed-point subspace of a group action consists of all vectors it fixes, and need not be one-dimensional.