290 UNIVERSITY OF COLORADO STUDIES 



[QT'']'=Q'^p'''+^>'^'p'°~n (3) 



Placing s=p and y=l 



If X be so chosen that 



x4-h = (mod p""-') 



QP* will be an operator of the order p and may be taken in 

 place of Q. 



The sub-group H, is generated by P and Q where 



PP"~ =1 Qp=l , 



with equation (1). 



Determination of G. 



G is generated by H, and some operator R of G which is not 

 contained in Hj. By the hypothesis of this class 



Rp=P''. G^R'^H,, a=0, 1, 2 . . . p-'. 

 Since H, is self-conjugate in G 



R-^PR = Q^P« (4) 



R-QR^Qbrap""-' (5)* 



[*Burnside, Art. 24.] 



By a repeated application of (4), (5) and (3) 



B-p P KP= q1^ /S P'-P [1+ -¥ ^ P"-'] 



D — a 





Hence 



b"— a" 



b— < 



y8 = (mod p) and 



\rQ „v 1 r" aP~^ aJ' b" ~l 



