172 CHAPTER VII. 



various reasons it would seem that the same result holds true when 

 m > 6, but no explicit investigation has yet been made. The devel- 

 opments in 191 193 are made on the assumption that this index 

 is 2. Moreover, if this conjecture prove false, very simple alterations 

 would be necessary in the treatment. 



186. We continue the investigations begun in 163 on the 

 senary group 6rl,2? whose substitutions leave absolutely invariant the 

 Pfaffian [1234], viz., 



-^4 = ^12 -*34 ^13 -*24 ~^~ -M4 ^23 



Denote by 6r 6 the group of all substitutions of determinant unity in 

 the GF[p n ~], p > 2, which leave F absolutely invariant. We will 

 prove that 6r 6 is holoedrically isomorphic with 0^(6,^"), where ^=1 

 or v according as p n = 4Z + 1 or 4? -j- 3. Hence 6r 6 has the order 

 ( 172) 



153) (p* n 



It will therefore follow from the theorem of 163 that 4,2 is a 

 subgroup of index two under the group 6r 6 . 



187. Let p n = 4:1 -f 1, so that 1 is the square of a mark i 

 belonging to the GF[p n ~\. We make the following transformation 

 of indices: 



-M2 fel ~J~ *2? " -MS '3 ~^~ ^4; -^14 = 55 "f~ ^Q) 



y t -t v ? - it Y it 



*- 34 = fel ~~ * 2 ; -*-24 ?3 *54? -t 23 ?5 ^$6' 



Then JP 4 takes the form 



Hence (r 6 is holoedrically isomorphic with 1 (6,p n ). By 164, the 

 following substitution of 6r 6 (leaving four of the indices fixed): 



belongs to the subgroup Cri } 2 if and only if r be a square in the 

 field. Expressed in the new indices, it has the form 



155) 



II - - Y ( - -*- 1 ) Is + (^ + T- 1 ) I 4 . 



For T an arbitrary mark =|= of the field, 155) may be written 

 156) 



