Crystal Classes and Systems

The word class is used to characterize diverse groupings of objects according to the properties they have in common. An equivalence class is made up of like (or conjugate) operations. Let a be a group of order m made up of the operations 9fc, 1 /c m. The equivalence class of the operation contains all the operations Ok The cosets of an invariant subgroup introduced 

A mirror plane and a twofold (or any even order) axis perpendicular to the plane generates an inversion center (Fig. 2U4or Fig. 2.13(c) with 0 = 90°). Two of these three elements generate the smallest non-cyclic (but Abelian) group this is a group of order 4 which comprises the operations E, A2J1, and I2. 

We will first derive the four types of point group that are composed uniquely of rotations (operations of the first type). These groups describe chiral objects or enantiomorphs. An enantiomorph and its mirror image are not superimposable. 

In the same way, a left hand is not superimposable on a right hand, and for any right-handed screw there exists an equivalent left-handed screw. 

From the considerations in the preceding pages (Section 2.5.2) we derive two types of rotation groups  

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