52 



4. The question may suggest itself: how can we be sure 

 that the results .will be different from each other if we combine 

 a certain group of the first order G with an operation of the 

 second order ft, and at another time with another operation of 

 this kind ft? Or in other words: when- will the resulting groups 

 of the second order Gj. and G 2 be different and when identical? 



The answer is given by a general theorem which can without diffi- 

 culty be proved to hold good in all cases: In combining a group of the 

 first order G with two different operations ft, and ft of the second 

 order, G and G^ will be identical if the rotation which would be the 

 result of combining ft and ft together, be already present among the 

 characteristic non-equivalent rotations of the initial group G; otherwise 

 G! and ~G Z ^ be really different from each other. J ) 



The significance of this theorem will at once be apparent, because 

 it gives a very simple criterion, whether, starting from a certain 

 rotatory group G we must expect to find a new symmetry-group 

 or a derived group identical with one already found on a former 

 occasion. 



5. Now that these general theorems are known, we can resume 

 our study of the groups C n again. From which precedes we may 

 now readily conclude that: 



a. If n be an odd number the symmetry of the group C n may be 

 also described as resulting from the existence of an axis of the first 

 order A n with the same period as A n , combined with a plane of 



symmetry perpendicular to A n . 



n 



b. If n be an even number, but odd, the symmetry of the group 



& 



l) The demonstration of this, theorem can be given quite simply if we 

 apply the method of Schoenflies here, who, following Jordan, Min- 

 nigerode and others, made use of the idea of the "multiplication of opera- 

 tions", after certain symbols for such operations are introduced, as they are 

 used in the theory of groups. 



Let Qi and 02 be the operations of the second order to be considered, e. g. inver- 

 sions or reflections. The "product" Qi . Q% is now of course equivalent to a 

 rotation A . Let us suppose it to be a rotation already present among those of 

 group G. If we multiply the equation Qj . Q2 = A by Qi, then, because Q 1 



is the identity, the result of this is: Q 2 = AQi. But AQi is an operation of 

 the second . order belonging to the new group GI , obtained from G by combi- 

 nation with Qi ; thus the equation : Qg = A Qi simply expresses that Qz is 

 also an operation of the second order characteristic of GI. But if so, GI would 

 also have been obtained if G were combined immediately with Qz\ instead 

 of with Qi ; i.e., GI and Gg, are identical. 



