174 



CHAPTER VII. 



Having every 3,' 4? contains their transformed 





the even substitution (is 6* 



by 



By 164, Note, contains the 



product Of; 2 Of; 4 and therefore also every Of// 

 are distinct. Hence contains 



?' CT 

 fell? 



where i, j, ~k, I 



and therefore every 0///0f ? 'f in which two of the subscripts are alike. 

 For the case p n = 5, i = 2, there is an additional generator, viz., E IM . 

 Expressing R 123 in the indices T^-, we obtain the substitution 



By inspection, this is the second compound of the following sub- 

 stitution of determinant unity with coefficients mod 5: 



The group therefore contains all the generators of OJ(6, p n ). 

 Since is of index 2 under O l (6 ) p n ') and 0{(6,^) n ) of index at 

 most 2 under 0^6,^) ( 182), it follows that = 0^(6,^). We 

 have therefore, by 163, the theorem: 



for p n = 4 1 -f- 1 , the group Oj[ (6, jp w ) /aos a maximal invariant 

 subgroup { I, T= CC 2 . . . C G ] of order two, the quotient- group being 

 holoedrically isomorphic witli the simple group LF(4:,p n ). 0{(6,p n ) is 

 of index two under the first orthogonal group O x (6, p n } and is extended 

 to it by any OfJ not a Q^. 



188. Let p n = 4^ + 3, so that 1 is a not -square in the 

 We make the following transformation of indices: 



r u =$,-s 



,, 



where a and /3 is a suitable set of solutions in the field of 

 159) a 2 +/3 2 =-l. 



