276 CHAPTER XII. 



For p = 2, Jc = 1, the group GQ of order 2 (1 + 2/ 1 ) is given by 

 the extension of 6rJ), formed of the substitutions S = I and ^, by 

 certain 2f extenders V$ (j = 1, . . ., 2/*) each of period 2. By 251, 

 all the substitutions of period 2 in GQ form one set of 1 -f- 2f con- 

 jugate substitutions. Setting F =$ 1? the substitutions of period 2 

 in GQ are F} (j = 0, 1, . . ., 2f~) and the remaining substitutions 

 F Vj = Uj are of period =f= 2. Hence no U is conjugate with a F. 

 The product Fj-C/J- cannot be a Z7; for the substitution of GQ which 

 transforms F/ into V Q transforms the product into F 7;=F, but 

 transforms the U into some U. Hence F/C/J- is of the form Vj" so 

 that Fo /;'/,- = F Z7/'. Hence every product CJ-CJ is a CJ-. TAe 

 substitutions U form a group GI . Since 7J-=FoF}, we have, for 

 every j, 2 



254) TT^Ti-lT 1 . 



For 7J- and C/J' arbitrary, there exists in the 6ri a 7j- such that 



a K 



^^(^^-^-^(Rr^Fo-^^o^Fo-^Q-^-^Fo^^^ 



The group Gi+%f of the Z7 J s is therefore commutative and contains 

 substitutions of period > 2. By 244, it is a cyclic subgroup of 

 6r s q:i. In view of 254) the group GQ is a dihedron 6r 2 (i+2/) based 

 on the cyclic 6ri + 2 f ( 245). These groups G$ have therefore been 

 enumerated in 246 and may be dropped from further consideration. 



253. For case [B], p > 2, n/lc is even and p k l = d. The 

 2d marks r\ are the square roots of the p k 1 marks K and hence 

 are the distinct powers of ^ = y^o; where 3C is a primitive root of 

 the GF[p k ~\. In particular, there is a mark 77 = }/ 1. 



Within (r^ there are exactly 1 -f fp k groups conjugate with the 



G p ^(pk_ 1 ). The latter contains p* conjugate cyclic 6rp*'_i and hence 

 in all p k substitutions T of period 2, each conjugate with 



x o, 



Under G Q of order Q = (1 + fp k )p*(p k - 1), this T is one of a 

 system of (1 + fp k )p k or (1 -j- fp k )p k conjugate substitutions T 



according as T is within GQ self -conjugate under the cyclic 6r^*'_i 

 or under a dihedron obtained by extending the former by a sub- 

 stitution TO which interchanges the elements <x>, ( 242, 246). In 



the respective cases there would be at least fp zk or (fp k 1) p k 



substitutions F^jF} ( j > 0) of period 2, necessarily satisfying the 

 relation 251), 



