THE HYPERORTHOGONAL AND RELATED LINEAR GROUPS. 137 



147. Lemma. If a substitution S of the group G- m , piS be commu- 

 tative with O a r \f , for certain values of a, then the following coefficients 

 of S must be zero, , . 



Krj, Ktjy Kjr, CC j{ (j = 1, . . ., f; J =|= r, t\ 



Among the conditions for the identity 80%? = 0%? S occur 



( - !>,,+ fa, = o, (B _ 1X> _ ^ a _ t = 0? 



(j = l, ...,w; j+r, t). 

 Hence the theorem follows if the determinant 



(a - 1) (a* 8 - 1) + ftp'* 1 = 2 - a - UP" + 0. 

 The equation 2 cc a?* = has p s solutions in the GF[p 2s ]-, indeed, 



a p*'= (2 - X= 2 - ^ 9 = a. 

 But for a arbitrary there exists a mark /S in the GF[p 2s ~\ such that 



Hence there are sets of solutions a, /J for which the above determinant 

 does not vanish, as well as sets for which it vanishes. 



Note. Another statement of our result is that S breaks up into 

 the product of a substitution affecting only r and % t by a substitution 

 affecting only g, (j = 1, . . ., m; j + r, t). 



148. We proceed to determine the structure of the group 6r m?pjS 

 of order Q m , p , s . For w = l ? the group is a commutative (cyclic) 

 group of order p* -f 1- For m = 2, its structure was determined 

 in 144. 



The substitutions of G- m , p , s of determinant D = 1 form an in- 

 variant subgroup H m ,p ?s of order Q m , P ,s/(p*+ 1). Indeed, we have 

 shown that D must be a root of 



120) 



Inversely, substitutions do exist in the group 6r Wi p )S having as deter- 

 minants every root of 120); for example, T^ t and its powers, where r 

 is a primitive root of 120). Hence the factors of composition of 

 G~m, P ,s are those of H,n, A , together with the prime factors of p* -f 1. 

 Supposing w5>3, let I be an invariant subgroup of H 7?i) ^ ?s con- 

 taining a substitution 



not of the form 



T: g- = T|, (i = 1, . . ., m) [X+ 1 = 1, T* = 1]. 



