GENERAL LINEAR HOMOGENEOUS GROUP. 83 



a like power of d qi $*, . . ., by a like power of <5 d = 1, respectively. 

 Since we have, for any mark i/, 



a composition -series of H is given by 



H, H qi , H qiqt , . . ., Hg^ . . . qi= I. 



In view of the theorem proven in 104 107, we may state the 

 complete 



Theorem. - - The factors of composition of GLH(m,p n } are 



Piy P'2j - - , Pk, Q/d(j) n - 1), q l9 q 2 , . . ., q h 

 except in the two cases (m,p n ) = (2, 2) and (2, 3), when the factors of 

 composition are 2, 3 and 2, 3, 2, 2, 2 respectively. 



104. Theorem. - - Excluding the above two cases, the group H is 

 a maximal self -conjugate subgroup of f. 



Suppose that f contains a self- conjugate subgroup J" which 

 contains all the substitutions of H and still further substitutions. 

 We will prove that, aside from the two exceptional cases mentioned, 

 J coincides with l~. 



By hypothesis, J" contains a substitution 



which is not in H and therefore does not multiply all the indices by 

 the same factor. Hence, by Corollary I of 100, S is not com- 

 mutative with every B r , s ,i (f, s= 1, 2, . . ., m; r =)= 5). Changing the 

 notation if necessary, we may suppose that S is not commutative 

 with J5i,g f x,'a substitution of determinant unity and therefore in the 

 group f. It therefore transforms the substitution S of the self -conjugate 

 subgroup J into a substitution belonging to J. Hence J contains the 

 product 



c __ 

 1 = b * j# 1? 2 , | 



which does not reduce to the identity J. In calculating this product, 

 let O be the linear function by which S~ 1 replaces 2 . Then T is 

 seen to have the form, in which the values of the fa need not be 

 determined: m 



T: ii-J^fc/t/1 -g,--A a <l> ( = 2, 3, . . ., m). 

 j*-i 



Suppose first that the a,-! are not all zero, say cr 21 =|= 0. For 

 m > 2, we introduce new indices rj { defined by the substitution V of 

 determinant unity, 



171 = gi, 7^ 2 = g,, ^ = i- - ^ J 2 (i = 3, 4, . . ., w). 



6* 



