Symmetry Groups 
Concepts
Before providing a formal definition of both a group and a point group, we
will give an example of a group, namely, the point group 6.
The earlier section on rotational symmetry
demonstrated that the sixfold rotation operator rotated a benzene
ring to equivalent positions when applied to the sixfold rotation axis.
The point group 6 contains the following symmetry operators:
which correspond to anticlockwise rotations of 60°, 120°, 180°, 240° (or 120°), 300° (or 60°), and 360° (or 0°), respectively, about the sixfold rotation axis. (Remember that the rotation axis is the symmetry element and not the symmetry operator). These 6 symmetry operators form a group according to the rules given below.
A symmetry group is a set of symmetry operators that are combined together according to a particular set of rules:
With reference to the point group 6,
the action of a 60° rotation followed by, say, a 240° rotation
is equivalent to a rotation of 300°.
In symbol form this is written as:
Again with reference to the point group 6, this is the rotation axis of order 1 which rotates an object by 360°.
Thus in the point group 6, the inverse operator for a positive 60° rotation (6^{+}) is a negative rotation (6^{}) by 60°.
A point group may be defined as any group of symmetry operators that leaves one or more points in space unmoved. If it is a single point, it is fixed in space by (a) the centre of inversion when the point group posseses rotaryinversion axes (e.g −1, −3, −4, −5, −6, −7, ..., −∞), or (b) the intersection of a twofold (2) and one or more axes (e.g. 2, 3, 4, 5, 6, 7, ..., ∞) or (c) the intersection of a mirror plane (m) and a rotation axis (e.g. 2, 3, 4, 5, 6, 7, ..., ∞) perpendicular to it. This single fixed point is called the origin. If it involves more than one point unmoved, then the point group is polar as discussed later under the subject of polar point groups.
Although there exists an infinite number of point groups, when crystallographic considerations are taken into account, i.e. rotation and rotaryinversion axes must be of order 1, 2, 3, 4, or 6, then the number of such groups is limited to just 32. The symmetry operators present in these groups are:
Rotation Axes  1  2  3  4  6 

RotaryInversion Axes  1  m (= 2)  3  4  6 
The 32 point groups consistent with the 7 crystal systems are known as the 3dimensional crystallographic point groups.
© Copyright 19952006.
Birkbeck College, University of London.

Author(s):
Jeremy Karl Cockcroft Huub Driessen David Moss 