GROUPS OF ORDER P" 287 



Hence 



ap 1 



^ p — = (mod p^) and 



a — 1 

 a- [ 1 + ^ A p— 3j + h/5p f = 1 + kp--^ ( mod p— *) 



From the last congruence 



a" ^ 1 (mod p-) where in > 5 and 

 a = 1 (mod p) 

 Substituting 1-fap for a in the second congruence we obtain 

 after a reduction 



(1 + ap)^— 1 



[a+h/3] p^ = k p""-' (mod p""-^) 

 ;hat 

 1 (mod p) 



ap 

 from which, making use of the fact that 

 (l+gp)"— 1 

 ap^ 

 we obtain 



(a+h)8)p^=0 (mod p"-'). 



From these last two congruences 



(a+h/3)p^=kp'"-^ (modp"-'). (4) 



Equation (2) is now replaced by 



Q— lpQ=:Q^Ppl+ap . (5) 



The group G is completely defined by (5) and (4) with the re- 

 lation Qp'^=P''p-, These relations may be presented in a simpler form 

 through a replacement of operators. 



From (3), (4), and (5) 



,S(S 1) „ p 8(8 1) , , S(S — 1) X(X — 1) 



k^yp-+ r^^-^^3^^-'^ x-y^kp°-3+5|^^[y- H^-l)^(2B-l) _ 



