Improper isometry

An even number of reflections in the same plane send each point from (x,y,z) back to itself, which is equivalent to an identity operation (a proper isometry), while an odd number of reflections in the same plane is equivalent to a single reflection (an improper isometry). The product of two reflections in separate planes is a proper isometry. We distinguish two cases (1) If the two planes are parallel and spaced by a distance d, the isometry is a translation by 2d in a direction perpendicular to the two planes (2) if the two planes intersect at an angle a, the isometry is a proper rotation (or, simply, rotation) through 2a about the line of intersection (the axis of rotation). Translations and rotations are continuous operations because motion can be controlled by continuous changes in the distance or angle between the two mirrors. 

The product of three reflections in separate planes is an improper isometry. We again distinguish two cases (1) Two reflections in parallel planes combined with a third reflection in a plane perpendicular to the other two results is a translation-reflection or glide reflection (2) two reflections in intersecting planes... 

In analogy to combinations of even and odd integers under addition, the product of n proper isometries is a proper isometry regardless of whether n is even or odd the product of n improper isometries is a proper isometry if n is even and an improper isometry if n is odd and the product of a proper isometry and an improper isometry is an improper isometry. 

Translations and rotations are particular isometries, i.e. the distances between any two points in space are maintained. As translations do not really matter in our situation, we restrict our attention to linear isometries, i.e. to isometries that fix the origin of 3D space. The reason is that we can embed a molecule in 3D space so that the origin of the space coincides with the molecule s barycenter, which is fixed under every symmetry operation. It turns out that linear isometries are compositions of rotations and reflections, orthogonal linear mappings where the representing matrix has determinant 1. The determinant of a proper rotation is +1, that of an improper rotation -1. Reflections are improper rotations. 

The crucial point was that the point group acts on both the skeleton and the set of substituents, such that consideration of the induced permutation does not suffice. For example, ft is the identity permutation induced by the reflection in the plane of the molecule, it is induced by em improper rotation which we cannot see by inspecting ft, we have to consider the isometry instead. 

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