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If *A _{1}* and

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 *A*^{n}_{1}).

Furthermore, if *A _{1}* represents a symmetry operation,

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 *A _{1}* 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.

**Copyright © 1980, 1998 International Union of
Crystallography**