Chemistry - Relationship between the symmetry number of a molecule as used in rotational spectroscopy and point group

This is not in general true

Consider molecules a point group not containing inversion symmetry, e.g. $C_2$ hydrogen peroxide

The $C_2$ group has only two elements, $E$ and $C_2$, and the $C_2$ rotation operation maps between two identical arrangements of atoms. Both the symmetry number and order of the group are 2.

Matrix representations

All rotations in 3D space can be represented by a orthogonal 3x3 matrix with determinant 1, and the composition of rotations about the origin can be represented by matrix multiplication of the transformation.

More formally: the group SO$(3)$ (3D rotations) is isomorphic to the group of orthogonal real 3x3 matrices with determinant 1 under multiplication.

The number of symmetry elements in a point group formed from rotations (and which hence preserve chirality) will be the same as the number which can be represented by such matrices and the same as the number of equivalent orientations of the molecule (the symmetry number) as they are all representation of the same thing - 3D rotations.

Inversions

The group of all transformations that keep the origin fixed, including reflections as well as rotations is O$(3)$ - or the product of SO$(3)$ with the set $\lbrace I,-I\rbrace$ or the inversion operation. It maps rotations onto improper rotations (which include reflections - improper rotation by 0 degrees) and hence all symmetries in 3D space.

All point groups with inversion symmetry, such as the 3 listed in the question, contains this $-I$ element. $-I$ will multiple all the elements of the rotation matrix by $-1$ and create an orthogonal 3x3 matrix with determinant -1. By composition with each of the rotations elements of the point group it creates another element - an inversion, improper rotation or reflection - doubling the size of the group without increasing the symmetry number.

Conclusion: chirality matters

This rule is true for non-chiral groups. You can invert the molecule to get another symmetric copy for every rotationally equivalent copy, so the order of the group is twice the symmetry number.

For chiral groups that will create the enantiomer which isn't symmetrically equivalent so the order of the group and symmetry number are the same.