CHAPTER III. 



Equivalent and Non-equivalent Operations. Definition of Group. 

 The Problem of the possible Classes of Symmetry as a Problem of 

 the Theory of Groups. The Axial Groups. Some general Theorems. 

 Cyclic, Dihedral and Endo spherical Groups. -- Symmetry -Centre 

 and Enantiomorphism. 



1. In this and the next chapter it is proposed to investigate, 

 what combinations of symmetry-properties are possible in the case 

 of finite figures and objects, and in what way the various types of 

 symmetrical figures can be classified. Attention was drawn in the 

 previous chapter to the important fact that not every combination 

 of symmetry-properties can exist, because the simultaneously existent 

 symmetry-properties of a certain figure are always connected by 

 special relations which allow of a mathematical treatment of the 

 problem considered. With reference to the previous chapters, the 

 reader will remember what has been said about the equivalence of 

 symmetrical operations. In the same sense we shall speak in future of 

 equivalent operations, as being those that bring a figure from its origi- 

 nal position into the same final position. Thus e. g. if a regular hexa- 

 gonal pyramid be rotated about its vertical axis through an angle of 

 60, it will coincide with itself, because it now occupies the same part 

 of space as before ; but notwithstanding this, its new position is different 

 from its initial one, and the same is the case if rotations around the 

 same axis are executed through angles of 120, 180, 240, and 300. If 

 rotated through 360, the figure arrives however ki the same position 

 in which it was originally. All these rotations, except the last ones, 

 therefore, are non- equivalent motions, quite independent of the fact 

 that by each of them the pyramid will be brought to self-coincidence. 



If a symmetrical figure possesses an axis of the first order with 



