LINEAR HOMOGENEOUS GROUP IN THE F[2] etc. 217 



The conditions that an arbitrary substitution S, for which c) hold, shall 

 be orthogonal are the same as the conditions that it shall be a special 

 Abelian substitution. 



6. The most general 2m-ary substitution commutative with the 

 special Abelian substitution M = MiM% . . . M m has the form 



The group in the GrF[p n ], p > 2, commutative with M is identical 

 with C of Ex. 5. 



7. Setting -1=/ 2 , XfEEfe + Ity, A tJ = a {j - Iy ih S of Ex. 6 

 becomes 



Z: HvJfAyJ, (<-!,...,). 



If 1 be a not -square in the 6rF[.p w ], we may pass, inversely, from 

 an arbitrary substitution in the G-F[p 2n ] to a substitution S in the 

 G-F[p n ~\ by equating the coefficients of J and /. X leaves invariant 

 the function 



Hence, if p n be of the form 4=1 + 1, the group C is simply isomorphic 



with the hyperorthogonal group G- mt p , n - If 1 be a square in the 



G-F[p n ], we introduce the further indices Y f ^& J^,-, 5,-y = a,-,- + Jy,^-, 

 when <S becomes 



m 



leaving in variant S* X/Y f -. Inversely, from every substitution Z t we 



1=1 



derive a substitution of the form 8. The group of "dualistic" substitu- 

 tions is simply isomorphic with GLH(m,p n ), since the BIJ are determined 

 in terms of the A's. 



8. The simple group A(&,p n ), p > 2, of order 



contains just two sets of conjugate substitutions of period 2. The one 

 set contains -~-(.p 2n -}- l)p* n substitutions conjugate with T],_ i. Those 

 of the other set are conjugate with M M 3 and are in number 

 ~p Sn (p^ n + l)(p n + 1) according to the form 4Z + 1 of # n . 



