GROUPS OF ORDER P^* 293 



r' -V ' T^ni— 3 





The condition that the lowest power of Q ' or R ' in | P ' | shall 

 be the p!? gives 



either z' or y' not = (mod p) and t?''^*" 



either z' ' or y' ' not ^ (mod p). 



Let the lowest power of R' in |P', Q' | be R'^" then j^'"V 



R'S'^Q'S'p'Sp'"-'. • 



Substitute the values of P ' , Q ' , and R' in terms of P, Q, and R. 



R«' '^' ' Q«"y' ' P^' '8' 'P""-^ + ''^^'g~^^ ay- ' z' '^'^ =B.^' ^' (Jf 



which gives 



px ' 8 ' p'"-'-(-x8p'°-* -\ ^-^ -^ay ' z ' p"*-^ 



s"z" =s'z' (mod p) and 



s' 'y' ' ^s'y' (mod p). 



Elimination of s' gives 



s' '[y'z' ' — y' 'z'] = (mod p). 



Since the lowest value of s ' ' is to be p, then 



y'z"— y"z'not = (modp) (15) 



This includes the conditions given above. 



Substitute in (12), (13), and (14) the values of P', Q', and 

 R' in terms of P, Q, and R, and reduce 



j^z Qy+/3xz'px[l+«z'p— ^ + a^^— ^^2=^p— n 



X Cx 1\ 



+ k^z'— ^^"2 — ^p'""^ -[-a(yz' — y'z)p'"-^+kxy'p°^-^ 



:=R"Qyp^[^+^'P"''l (16) 



j^z^y+^xz" pX[l+az-p-°-^ + a^ ^ ^\ ~ ^ P""^ 



