53 



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

 that the results will be different if we combine a certain group 

 of the first order G with an operation of the second order Q lt 

 and at another time with another operation of this kind (X>? 

 Or in other words: when will the resulting groups of the second 

 order G 1 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 Q ly and Q 2 of the second 

 order, G 1 and G 2 will be identical if the rotation which would be the 

 result of combining Q^ and Q 2 together, be already present among the 

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

 G 1 and G 2 will 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 established, we can 

 resume our study of the groups C n again. From what 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 . 



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



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

 apply the method of Schoenflies, who, following Jordan, Minnigerode 

 and others, made use of the idea of the "multiplication of operations", after 

 certain symbols for such operations are introduced, as they are used in the 

 theory of groups. 



Let Qj and Q 2 be the operations of the second order to be considered, e. g. inver- 

 sions or reflections. The "product" Q^ . Q 2 is now of course equivalent to a 

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



A 



group G. If we multiply the equation Q 1 . Q 2 = A by Q v then since >i 

 means the identity, the result is: Q z AQ r But AQ l is an operation of the 

 second order belonging to the new group Gj, obtained from G by combination 

 with Qjj thus the equation: Q z = AQ l simply expresses that Q 2 is also an 

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

 have been obtained if G were combined immediately with Q^ t instead of 

 with Qjj i.e.: Gj and G 2 are identical. 



