Some examples of symmetry groups

To demonstrate group theory in action and its close relationship to the intuitive notion of symmetry, some elementary groups of importance in solid-state science will be considered. 

This is a symmetry group that describes the rotations of a regular polygon with n directed sides, e.g. 

The elements of the group are rotations through an angle 2-Kj/n (j = 0,1, 2. n — 1) about an axis through the centre and normal to the plane of the figure. Calling these rotations C]n it is clear that a rotation CJn is equivalent to repeating the smallest non-trivial rotation C j times, i.e. 

Mathematically this group is known as a cyclic group of order n, generated by the element c, and consisting of the elements e, c, c2,. ..,cn-1. The multiplication table of C3 is 

Each element appears once only in each row and in each column. 

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