186 CHAPTER VII. 



It follows that G" is holoedrically isomorphic with OJ-(6, p n ~) or 

 Oi(Q,p n } according as p n = 41 + 1 or 4Z + 3. But, for p > 2, the 



order of the second compound H^ of H' is h* and therefore 



equals that of 0^(6, p n ~). Hence G", H^ and 0^(6,j) n ) are holoedric- 

 ally isomorphic. 



By 132, we pass from H 1 to the quotient -group .iELi(4, p* n ) 

 by making the substitutions T K (106) correspond to the identity. The 

 corresponding substitutions of H^ are the identity I if p n = 4? -f 1, 

 but are J and the substitution T changing the signs of the six indices 

 if p n =4:l + 3. Hence Oi(6,y) is holoedrically isomorphic with 

 HA(4,p* n ) if #=4Z + 1; while, for #=4Z + 3, Oi(6,p") has the 

 maximal invariant subgroup { J, Q C 2 C 3 (7 4 (7 5 C 6 } of order 2, the quotient- 

 group being isomorphic with HA(4, p 271 ). 



191. We proceed to determine the structure of the orthogonal 

 subgroups 0^(m, p n \ m 5> 7. Every w-ary linear homogeneous sub- 

 stitution is commutative with 



0=0,0,... C m : 65 - - fe ( - 1, . . ., ). 



belongs to the group Oj[(i,jp*) only when m is even and ^,= 1 

 (see 185). Suppose that Ol(m,p n ) has a self - conjugate subgroup G 

 containing a substitution 8 neither the identity I nor C: 



Suppose first that /S reduces to the form 



171) i!-8,- (t-l,...,i) 



where f t -=l. Then $ is merely a product of an even number of 

 the d, in which certain ones as C* are lacking since S =j= (7. If ft = v 

 and therefore m even by hypothesis, we may suppose that both C m 

 and C k (k <m) are lacking, since CiC m does not belong to Ol(m,p n ). 

 But if S^dCjCrCs..., its transformed by T ti T i1s (always in the 

 main group) gives 8' = C k CjC r C s . . ., so that G contains the product 



From it we obtain in G the substitution C^Cg and are thus led 

 to the case treated in 193. 



Suppose, on the contrary, that 8 is not of the form 171). We 

 may assume that 12 , 13 , . . ., a im are not all zero. In fact, either 8 

 or its reciprocal will have at least one ,-y =[= in which i < j. 

 Transforming the one or the other by T^Tu, if j < m, we obtain a 

 substitution in G which replaces ^ by 



