= Solution
For a vector $x$, its <isotropy group> is the <stabilizer subgroup>
$$
\boxed{G_x=\{g\in G:R_G(g)x=x\}.}
$$
For a <subgroup> $H\le G$, its <fixed-point subspace of a group action> is
$$
\boxed{\operatorname{Fix}(H)=\{x\in\mathbb R^n:R_G(h)x=x\text{ for every }h\in H\}.}
$$
It 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.
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