Symmetry operations, the point group

Summarizing, we obtain the following important notions and results that allow the precise description of symmetry operations of molecules and their point groups  

2 Definition (Orthogonal group, point group, symmetry elements) 

These mappings of 3D space are the mappings represented by orthogonal matrices, i.e. they are elements of the group of matrices representing linear isometries, also denoted by O3, for sake of simplicity of notation  

Basically, an isometry of Euclidean space can be represented, with respect to a suitable basis (bo, bj, b2), by a matrix of one of the forms 

The entry 1 in the upper left hand corner is +1 if the matrix represents a proper rotation, while it is -1 in the case of an improper rotation, i.e. a rotation by the angle rj combined with a reflection. Suitable basis (bg, bj, b2) means that bg gives the rotation axis (bo is a point of this axis), while bj,b2 span the reflection plane. If we choose t] = 0° we obtain the particular matrix 

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Cyclic groups contain only operations derived from the repeated application of a single rotational symmetry operation. The point group is C if the repeated operation is a simple rotation, and we have the point group S if it is an improper rotation axis. In both cases the subscript denotes the order of the axis. 

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