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If A1 and A2 are two matrices representing a symmetry operation, it is not difficult to demonstrate that the product matrix also represents a symmetry operation. In fact, since A1tGA1 = G and A2tGA2 = G we have:
This result obviously holds not only for the product of two matrices , but also for the product of several matrices (a special case of this is An1).
Furthermore, if A1 represents a symmetry operation, A-11 also does: in fact from relation , pre- and post-multiplying both members by (At1)-1 and by (A1)-1 respectively, and keeping in mind that (At1)-1 = (A1-1)t we obtain:
Finally it is obvious that matrix 1 represents a symmetry operation (identity) no matter what the base system defined by G may be. In this way we have demonstrated that all group theory postulates are applicable to the symmetry operations. Therefore the symmetry operations are the elements of a group, called a symmetry group. Since all symmetry operations A1 leave a point with coordinates (0, 0, 0) unchanged, (i.e. all the symmetry elements pass through that point) these symmetry groups are called point groups.
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