218 



CHAPTER IX. 



9. The group of all quaternary linear homogeneous substitutions in 

 the G-F[p n ~\ which leave absolutely invariant the functions ^^ + | 2 ^ 2 

 and Jj_ -f- ^ has a subgroup of index 4 holoedrically isomorphic with 

 LF(2,p). 



10. The squares of the substitutions of the first orthogonal group 

 1 (m, p n ) generate the subgroup 0[(m, p n ) of 181. 



11. To the subgroup E^ p n of E^ p n corresponds, for p > 2, the 

 subgroup OJ(4, p n ) of O v (,p n ) defined in 181. 



12. In order that AI? + Aai? shall be capable of transforma- 

 tion into |ii(i + |f ) by a binary linear substitution with coefficients 

 in the G-F[p 2s ], it is necessary and sufficient that the ratio ^/^ shall 

 belong to the GF[p*~\. 



CHAPTER IX. 



LINEAR GROUPS WITH CERTAIN INVARIANTS 

 OF DEGREE q > 2. 



211. Consider the group 6r 3 of substitutions in an arbitrary field 



S 



+ Pays + vnefi (t 1, . . ., r) 



which leave absolutely invariant the function of degree q = 3 



It will be convenient to employ a symbol, analogous to a determinant, 



A B C t 



^Afic -f Ayb -{-Bac + Bya + Cab + (7/3a. 



The conditions that S shall leave O 3 absolutely invariant are then 

 215) 



