280 CHAPTER XII. 



into itself and hence belong to Gr*M(^). The latter thus forms one 

 of a system of M(s)/2M(p k ) conjugate subgroups. 



It remains to determine the number of systems of conjugate 

 subgroups of these two types; indeed, in 251, there entered the 

 transformer PT/^TI which belongs to GM( S ) if and only if A is a 

 square in the G-F\_p ti ]. For p = 2, A is necessarily a square; for 

 p > 2, nfk odd, A may be chosen as a square, since every additive- 

 group [A 1; . . ., AJ with the multiplier GF[p k ~\ has half of its non- 

 vanishing marks squares in the GF[p n ]. In these two cases there 

 is evidently but one system of conjugate subgroups G M ( P Je ) of GM( S ). 

 For p > 2, n/Tc even, all the marks of [A 1; . . ., A*] are squares or all 

 are not -squares in the 6rF|j?*]; indeed, they are all obtained from a 

 single one by multiplication by the p n marks of the multiplier GF[p k ~] 

 and the latter are all squares in the GF[p n ]. In this case there are 

 consequently two systems of conjugate subgroups GM( P *) and two 

 systems of conjugate G^M(p k ) 7 the systems of each type being inter- 

 changed upon transformation by P-i/7, belonging to GZM(S), where v 

 is any not- square in the GF[p n ~\. Hence there are (2, 1; 1) systems 

 of conjugate G M ( P k ) and (2, 0; 0) systems of conjugate G^ M(p k) within G M ( S y 



256. Subgroups of G-M( S ) containing no operators of period p. 

 Every substitution of such a subgroup GQ lies in and determines a 

 largest cyclic subgroup G- d of 6r$ ( 242 243). Two such groups 

 G- d have only the identity in common. According as G d is self- 

 conjugate within G$ only under itself or under a dihedron G% d based 

 on G d} it is one of a system of Q/d or Q/2d conjugate subgroups 

 of GQ. Let r denote the number of such systems. The enumeration 

 of the substitutions of G leads to the relations 



256) Q - 1 +,.-l) (#_ i or 2 ) 





257) 



If two non- conjugate cyclic G d ., G d . of odd order are present 

 in (TQ, there are at least dj groups in the system determined by G d{9 

 viz., the transformed of the latter by the operators of G djJ and vice 

 versa, so that 



258) Q^di (dj - 1) + dj (d t - 1) + 1. 



Solving 256) for 1/Q, we get 



