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A point group is a group of spatial symmetry operations fixing at least one common point. Choosing that point as the origin represents its operations by orthogonal matrices, so the point group is a subgroup of the orthogonal group. A pure nonzero translation cannot belong to it.^[ref]

A crystallographic point group additionally preserves a full-rank Euclidean lattice and is finite. For a crystallographic space group, the group of orthogonal linear parts is its quotient by the translation subgroup; it preserves the translation Euclidean lattice. In two dimensions the crystallographic restriction theorem allows only rotation orders $1,2,3,4,6$.

The broader definition is needed for molecular point groups. A regular planar pentagon has a fivefold rotation, so its finite point group is not crystallographic. An ideal centrosymmetric linear molecule has the infinite D infinity h molecular point group, including rotations through arbitrary angles about its axis. Thus neither finiteness nor lattice preservation belongs to the unrestricted definition.