GROUPS OF ORDER P'^ 291 



From the last congruence 



aPEEl(raod p*"-^) and 



a = l(mod p™"*). 

 Equation (4) is now replaced by 



R-lpR^Q^pl+^P""" . (6) 



From (5), (6) and (3) with (2) 



R-pQRp=p- 'pQP'p^Qb" p^^^ap'"-^=Q. 



Hence 



b'' = l(mod p) and 



bP-1 

 a^^— j=0(modp). 



From the first congruence 



b=l. 

 Equation (5) now becomes 



R-'QR=:Qpap'"~' (7) 



and equation (6) becomes 



R-TR=:Q^pl+ «P"~'. (8 ) 



This last comes by the substitution of l+ap™~* for a and 1 for b in 

 the congruence determining a. 



From (3), (7) and (8) 



R-p^R^^Q'^^y p^ti+«yp'"-^+«/3 ^-^ p-'-l 



+k^y^-^^p-% (9) 



R-y P='Ry=Q-paxyp'"-^ (10) 



From (2), (9) and (10) 



S(8— 1) ^ ^ S(8— 1) 



[R^Q^P7==R^^Q^y+^2 ^^^P^tB+— 2— ^-P""-^ 



