 Solution

A screw axis combines rotation about a line with translation parallel to that line. An $n_p$ screw rotates through $2\pi /n$ and advances by $p/n$ of the lattice repeat along the axis; a $2_1$ screw advances by half a repeat after a half-turn. The combined operation, rather than either component separately, must map every atomic species onto equivalent sites.

A glide plane combines reflection in a plane with a fractional translation parallel to it. Reflecting alone need not preserve the crystal. A crystallographic point symmetry is an isometry with a fixed point, such as a rotation, reflection, inversion, or rotoinversion. Point operations have no irreducible translational advance: a screw or a glide with nonzero fractional advance cannot be described merely as a rotation or mirror at a different origin. A crystal's point group records the rotational parts of its space-group operations, whereas the full space group retains their translations.