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Symmetry groups

If A₁ and A₂ are two matrices representing a symmetry operation, it is not difficult to demonstrate that the product matrix $A = A_1 \cdot A_2$ also represents a symmetry operation. In fact, since A₁^tGA₁ = G and A₂^tGA₂ = G we have:

$$(A_1 \cdot A_2)^t \cdot G \cdot (A_1 \cdot A_2) = A^t_2A^t_1 \cdot G \cdot A_1A_2 = A^t_2GA_2 = G.$$

This result obviously holds not only for the product of two matrices $A_1 \cdot A_2$, but also for the product of several matrices $A_1\cdot A_2\cdot A_3 \cdots$ (a special case of this is Aⁿ₁).

Furthermore, if A₁ represents a symmetry operation, A^-1₁ also does: in fact from relation $A^t_1 \cdot G \cdot A_1 = G$, pre- and post-multiplying both members by (A^t₁)^-1 and by (A₁)^-1 respectively, and keeping in mind that (A^t₁)^-1 = (A₁^-1)^t we obtain:

$$G = (A_1^{-1})^t\cdot G\cdot (A_1^{-1}).$$

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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