67 



rent. But D% need not be considered, because we found it identical 

 with D^', so it is once more confirmed, that D% and D% are 

 really different from each other. 



Similarly S v and / are together equivalent to a rotation through 

 1 80 round a binary axis perpendicular to S v . As S v passes through 

 a binary axis of D n> the resulting binary axis will be perpendicular 

 to one of the binary axes characteristic for D n . If n is even, such 

 an axis perpendicular to one of the other binary axes will already 

 be found among those of D n . If n is odd, this would not be the case. 

 Therefore, only if n is an odd number, will the groups .D% and 

 Dn be different; but D% being for all values of n the same as 

 Dn, it is only demonstrated here once more that Dn and D n 

 are wholly identical for even numbers n, and only different if n is an 

 odd number. But, as we have seen, for n = odd, D n is identical with 

 Dn already deduced. 



Finally we have to consider the last possible combination: that 

 of S D and /. The result of both is a rotation through 180 round a 

 binary axis perpendicular to S D . Now, if n. is odd, such binary axes 

 will already have been found among those of D n ; if, however, n be 

 even, it will be a new r one with respect to the binary axes of D n . 

 From this it follows, that only when n even, can the deduced groups 

 Dn and D n be different from each other. But if n is an even number, 

 D^ will be always the same as D^', so that our result amounts 

 to saying that for even n the groups D% and )Jjf will differ, - 

 just as we have already stated. Since, because for odd n, Dn is 

 certainly different from Dn, it is here once more demonstrated 

 that in all cases Dn and Dn must be different from each other. 



In this connection it is of importance to draw attention to a 

 special property of the axis A n , if such a dihedron-group D n is made 

 into one of the second order by adding a diagonal mirror-plane 

 S D to it. It can easily be proved by means of group- theoretical ar- 

 gumentations, that in this case the axis A n is transformed at the 



same time into an axis A 2n f the second order l ) with a period of . 



l ) For the general and simple demonstration of this theorem, the same 

 symbols for the "multiplication" of operations of the first and second order 

 can be used as we drew attention to previously. Let S D be the diagonal 

 plane bisecting the angle between two successive binary axes of D n , and let 

 A 2 be a rotation through 180 round such an axis; S H and 5^ may be positions 

 of planes of reflection, as we have defined them in 9 of this chapter. 

 Then we have: A 2 = S^.S^, and therefore A Z .SD = SH.sr.sD. NOW 



