94 



CHAPTER II. 



and is therefore the identity. Hence S is a product of substitutions 

 of the given types. Since the latter are all of determinant unity , so 

 is also the general substitution S of the group. 



115. Theorem. - The order SA[2m, p n ~\ of the special Abelian 

 group equals 



8) 2n 



There are (j? n ) 2m 1 sets of values of a lj7 y\j (j = 1, . . ., w), not all 

 zero, which give distinct functions c^. In the function o? 2 , d n = 1 

 while /3n, jSiy, diy (j = 2, . . ., m) are arbitrary in the field. Hence o> 2 

 may be chosen in (# n ) 2m ~ * ways. We have therefore the recursion 

 formula 



- 2, #]. 



116. Theorem. For p > 2, the factors of composition of 



',j p n ) are SA[2m, p n ~] and 2, the case p n =3, m *=* 1 being 



exceptional. 1 ) 



Every substitution of Gr ~ SA(2m, p n ) is commutative with 



The group K = { J, T) of order 2 is therefore self- conjugate under Gr. 

 In order to show that K is the maximal self- conjugate subgroup 

 of 6r, we prove that a self -conjugate subgroup J of Gr, which 

 contains K without being identical with K, must coincide with Gr. 

 Let S, given by 75), be a substitution of J not in K. Then J 

 contains the products 



S Li t i SLij, S L'i,i SL' f) i (i = 1, . . ., m) 



where I is a fixed mark =|= 0. Suppose first that all of these 

 products reduce to the identity. Then, for example, S is commutative 



with both LI,* and L 

 100, S has the form 



so that, by the proof of Corollary I of 



1) For wz. = 1, /SJL(2m,p) is identical with the group of all binary linear 

 homogeneous substitutions of determinant unity. Its factors of composition are 

 therefore given by the theorem of 103. 



