286 UNIVERSITY OF COLORADO STUDIES 



groups of order p", ra=l, 2, 3, 4, being given by Burnside pp. 



87-88.] 



Class I. 



The group G is generated by P and Q since the operators Q^ P** 

 /3=0,1,2 . . . p'— 1, a=0,l,2 . . . p^^lf, are p™ in number and 

 are all different. We have the relation Q''^=P*'''^ In G there is a 

 sub-group H, of order p""' which contains |P| self -con jugately, and 

 H, is self-conjugate in G. 



[Burnside Theory of Groups, Art. 54, p. 64.] 



H, is generated by P and some other operator Q^ P" of G. 



Then Q^ is contained in H, and Hj is the sub-group |P, Qp( and the 



operations of H, are of the form Q^^P**; /3=0, 1, 2, . . . p"'; 

 a=0,l,2, . . . p-zf. 



From this we have the two equations 



Q-ppQp=F+''p"-' (1)* 



Q-TQ=Q^PP« (2) 



[♦Burnside Theory of Groups, Art. 56. ) 



Determination of the sub-group H,. 



By a repeated application of (1) we obtain 



Q-y,. px Q.„ ^ pxLl+kp-^j''^ p x[l+kyp-'] 



where m >^ 4 •* 



and from this it follows that 



j-Qy,.pxj«^Qs>ppx[8+ky— ^— p"'-'] ^3^ 



Determination of G. 



It now follows from (2) and (3) 



Q-^PQ-^Q^P-^iTP " LA+^-^::^p J 



^pl+kp'"-' 



