Note 6.

The term group comes from the mathematical theory of Groups. In our case a point group means a certain set of point symmetry operations. Between the individual elements of such a set (thus between the relevant point symmetry operations belonging to that set) one can imagine an operation to be possible, namely the execution of one such symmetry operation folowed by a second one executed on the result of the first one. For example we subject a certain object first to a rotation involving a certain angle, resulting in the object to coincide with itself, and now we subject this result to a second symmetry operation, say a rotation or a reflection, resulting in the object again coinciding with itself. This means that the combined operation is itself a symmetry operation, that should be identical to one of the symmetry operations of the above mentioned set. If any combination of symmetry operations of the set results in an operation that is already represented in that set, then such a set is called a group. Such a group is accordingly a more or less small mathematical system : It consists of elements which can be combined by means of a certain operator, in our case this operator is : the execution of two or more symmetry operations after each other. All such consecutively executed symmetry operations result in a symmetry operation already listed in the set. We say that the set is closed under the operation of combining two or more symmetry operations.

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