GROUPS OF ORDER P"" 289 



The type group is generated by the independent operators P and 

 Q, which are subject to the relations 



pP =1 Qp2=l 



Q-TQ^Q^pl+P""* 



By an analogous replacement of operators in the other cases we 

 obtain the type groups 



2' Q-TQ=Qppl+P°'~' 



3" Q-TQ=Qpp 



4« Q-TQzrrpl+P""^ 



50 Q-'PQ=pl+P"~' 



6° Q-TQ=P 



The last three of the above types contain |P| self -con jugately, 

 while the last is the Abelian group of the type (m — 2, 2). 



Class II. 



There is in G a sub-group H, of order p"'"^ which contains |Pj 

 self -con Jugately. (Burnside, Art. 54, p. 64.) 



This is generated by P and some operator Q. By the hypothe- 

 sis of this class Q"" is contained in \V\. Qp=P^''. 



All the operators of H, are given by 



Q^P*^ a=^0, 1, 2, 3 . . . p-z?; ^=0, 1, 2, . . . p-'. 

 From this 



Q-TQ=pl+kp'^"' (1)* 



[*Burnside Theory of Groups, Art. 56.] 

 Determination of H,. 

 From (1) 



Q-P^Q^=P^[l+kp""']y=P^[l+kyP"~'] (2) 



where m > 4, 

 and from this it follows 



