= Site-symmetry group
{title2=$G_{\mathbf r}$}
The \b[site-symmetry group] of a point $\mathbf r$ in a <crystal structure> is the <subgroup> of its <space group> $G$ that fixes that point:
$$
G_{\mathbf r}=\{g\in G:g\mathbf r=\mathbf r\}.
$$
It is the <stabilizer subgroup> for the action on points. Although the <space group> contains infinitely many lattice translations, the site-symmetry group is finite and is isomorphic to a subgroup of the corresponding <crystallographic point group>.<Site symmetry and Wyckoff positions in the IUCr Online Dictionary of Crystallography>[Site symmetry, Definition]{ref}
For an original example, take a lattice with inversion symmetry and no other point operations, the space-group type conventionally denoted $P\bar1$. At the origin, both the identity and inversion $\mathbf r\mapsto-\mathbf r$ fix the site, giving a group of order2. A generic site has only the identity in its site-symmetry group; inversion instead produces a distinct site at $-\mathbf r$.
The equality in the definition fixes the actual point. At $\mathbf r=\mathbf a/2$, inversion alone moves it to $-\mathbf a/2$. Inversion followed by translation by $\mathbf a$ fixes it. Reducing fractional coordinates modulo integers can conceal that translation, but the full <space group> operation retains it.
The <crystallographic site multiplicity> consequently satisfies
$$
m=\frac{m_{\mathrm{general}}}{|G_{\mathbf r}|}.
$$
In this primitive inversion example the general multiplicity is2, so the origin has multiplicity1 while a generic site has multiplicity2. This is why high site symmetry reduces the number of distinct sites generated in a cell.<Atom-site symmetry and disorder in the IUCr core CIF dictionary>[`_atom_site_site_symmetry_multiplicity`, Definition]{ref}
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