266 CHAPTER XH. 



From the latter two follow tlie relations (holding for any integer r) 



The cyclic subgroup G- k generated by A is therefore self - conjugate 

 under G^. The latter is said to have the cyclic base G k . The ~k operators 



BA* (*-0, 1,..., fc-1) 



are of period two. For & odd, they are aU conjugate under G^ 

 since B transforms BA into BA~ 1 = BA k ~ 1 , which belongs to the 

 series B, BA 2 , BA*, . . . For Jc even, they form two sets of con- 

 jugate operators 



., 



BA, BA S , 



According as Jc is odd or even, they generate cyclic groups 6r 2 forming 

 one set or two sets of conjugate subgroups. 



For every divisor d of Jc, Gk contains a single cyclic subgroup G dy 

 which is formed by the operators 



A s , A 2 *, A^,...,A dd = I (d = Jc/d). 



If ^ be a given one of the integers 1, 2, . . ., d, the following d operators 



extend the cyclic group G d to the same dihedron G% d . There are 

 exactly d such dihedron -groups. If Jc be odd, these G$ d are all 

 conjugate under G% j. If d be odd, but & be even, the exponents k w, 

 ^ + d, [i + 2$, are alternately even and odd, so that each G% d 

 contains operators of both of the sets 249)*, the groups G% d are 

 therefore all conjugate under 6r 2 jt. If d be even and hence & even, 

 the exponents are all even or all odd, so that the operators all belong 

 to a single one of the two sets 249); the groups G% d thus belong 

 to two distinct systems of conjugate subgroups of Gr 2 k. 



If d > 2, Gz d has a single cyclic G d and G^ a single cyclic Gr k , 

 so that the above process furnishes every dihedron subgroup G^ d 

 of 6r2jfc. The theorem stated below therefore follows if d > 2. 



We consider next the case d = 2, k even and > 2. The only 

 operators of period two in G^ are then A*/ 2 and 



BA (^ = 0, !,...,& -1). 



Hence any dihedron 6r 4 must contain two operators BA r , BA S (r =f= 5) 

 and therefore their product BA r BA s ^A s ~ r . Hence every 6r 4 must 

 contain A k / 2 and may therefore be based on the subgroup 6r 2 of G k . 

 The theorem then follows as before. The jfc/2 possible groups 6r 4 

 in &2k are given by the formula 



{I, A*'*, BA', BA'+*i*\ (r~ 0,1,... ,-!- 



