THE HYPERORTHOGONAL AND RELATED LINEAR GROUPS. 131 



But this function is invariant under the transposition (i^i/,) and hence 

 cp r must have heen invariant under a substitution in which a^ =j= 0. 



It follows that '". 



0,-j = (i = 1, . . ., m^ j = m { + 1, . . ., m) 



in every substitution leaving cp r invariant. Considering the form of 



the reciprocal, we have 



Kji = (i = 1, . . ., m^ j = w t -f 1, . . ., m). 



Hence every substitution leaving (p r invariant is the product of two 

 commutative substitutions , the one affecting the indices % lf . . ., | Wi 

 only and leaving invariant 



and the other affecting only | TO| +i, . . ., m and leaving invariant 



Proceeding with the latter substitutions in the same manner, it 

 follows that the structure of the group in the GF[p n ~\ leaving <t> r 

 invariant results immediately from the structures of various linear 

 groups in the GF[p* ns ] denned by invariants of the type O. But 

 the relations 119) for substitutions of the latter groups become 



a fj S=tt ij (t, j = l,...,w). 



Hence there is no limitation imposed in assuming that the field to 

 which the substitutions belong is the 6rF[jp 2 *]. 



143. We designate by G m , P , s the group of all linear homogeneous 

 w-ary substitutions in the GF[p 2l< ] which leave <t> invariant. For 

 p > 2, those of its substitutions whose coefficients belong to the 

 GF[p s ] constitute the first orthogonal group 1 ) in the G-F[p > '] on m 

 indices. Indeed, relations 117) and 118), for A,-=l, then become 



The group G^p^, having the orthogonal group as a subgroup, will 

 be called the hyperorthogonal group in the GF[p* s ] on m indices. 

 We proceed to determine its structure, treating first the case m = 2. 



144. Theorem. If p*> 3, the group of the substitutions of G^ p , 9 

 of determinant unity has a maximal invariant subgroup of order 1 or 2 

 according as p = 2 or p > 2; the quotient -group is LF(2j p'). 



1) See Chapter VJI, 171. For p = 2, see Ex. 4 of 210. 



9* 



