138 CHAPTER V. 



With the single exception m = 3, p* = 2, when H 3i 2i i is of order 72, 

 we shall prove that I coincides with H. Therefore the substitutions 

 T form a cyclic group of order d, the greatest common divisor of m 

 and p* -f 1, which is the maximal invariant subgroup of H m? p , s . Hence 



the quotient -group gives a simple group of order //"^Vr We shall 

 designate it by the symbol H0(m, p 2 *). 



149. Theorem. There exists in the group I a substitution replacing 

 ^ 'by x^-J- <yJ 2 and not reducing to the identity. 



Suppose that a 18 =j= 0, for example. Transforming S by 0|?, we 

 obtain a substitution /S" replacing ^ by 



.7=4 



To make the coefficient of 3 zero, we have the conditions 



The condition for u is therefore 



Unless ^ s + 1 -f- fj +1 = 0, there exists a solution t u in the GF[p* s ~] 

 of this relation; indeed, the value of t it^ s + 1 belongs to the Q-F\_yp\ 

 and is therefore the ,(jp*+l)*' power of a quantity t a in the 6rJF[jp 2 *]. 

 It follows that we can assume that the only coefficients a 1; - ( J > 1) 

 which do not vanish are cr 12 , . . ., ai TOl and that, if % > 2, they have 

 the property that 



128) f ;+i++i_o (j, *-.,.--,i5 J-|-*> 



If m 1 = 2, the theorem is proven. If m 1 > 3, the terms in 128) 

 must all be equal and therefore zero unless p = 2. Supposing first 

 that p ={= 2, our theorem is proven unless m 1 = 3, when we have 



129) ag+'-l, g + '+g+'~0, -<) 0-4,..., ). 

 In the latter case we may assume that not both 



+i+ Bf M-i_0 (* = 2, 3); 



for, if so, f2 +1 = tt f3 +1 and hence each is zero by 129), since p =j= 2. 

 For definiteness, let 



If the left member be unity, then 12 = by 129) and the theorem 

 is proven. Suppose therefore that the left member is neither zero 

 nor unity and consider the substitution 



