The Mathematical Basis of Groups

Consider any point in the upper right x, y) quadrant of a standard 2-D graph. The value of z in this case is zero, so the 3-D coordinate set is (x, y, 0). Operating on this point with the inversion symmetry operation  

Is the point (—x, —y, 0) a point on the line y = x Yes, it is, for any value ofx. Such points are in the lower left quadrant. Therefore, this equation contains a center of inversion. To convince yourself of this, plot the graph and repeat the example. 

We have established two things about symmetry operations. First, they are operators and expressed mathematically in terms of a 3 X 3 matrix for operations on a point in 3-D space. Second, we have stated that only certain collections of symmetry elements, called point groups, are possible for real objects. 

The area of mathematics that deals with symmetry and point groups is called group theory. A group is a certain collection of operations that satisfies the following conditions  

This means that if you have three symmetry operations labeled A, B, and C, the combinations AB)C and A BC) must yield the same overall effect. 

---