THE HYPERORTHOGONAL AND RELATED LINEAR GROUPS. 127 



from which r) = 1. We may therefore limit our discussion to the 

 case in which r is prime to p. 



In order that the linear substitution on m > 1 indices 



shall leave r formally invariant, the following conditions upon its 

 coefficients must be satisfied 1 ): 



m 

 -j i ~ \ ^^"1 i r i / -i \ 



ILUJ > AiCCij == AJ {j = i, . . ., m) 



1=1 



116) rlr ., r ' ,, y, vn ?>... a,7. - o, 



holding for every partition of r into s integral parts 



r = r x -f r 2 + f- r,, m > s > 1, 



while for each partition J 1? j 2 . . ., j 5 may take every combination 

 of s distinct integers chosen from 1, 2, . . ., m. 



If r be not divisible by p, the inverse of S is 



Indeed, the product fi^/S replaces | & by 



m / OT \ 



IT 2 1.2 * a '*" J a ' ) & ^ ^' 



* ^- = 1 V; = i / 



upon applying 115) and 116) for r = r 1, r 2 = 1. 



140. Theorem. - - I/* r > 2, /" r be not a multiple of p, and if 

 r 1 &e wo^ a power of p, the only linear homogeneous substitutions in 

 the G-F[p n ] which leave r invariant are those which merely permute 

 the terms AII, . . ., l m ^ m amongst themselves. 



Consider for r > 2 the following equations of the set 116), in 

 which ^ and j 2 denote two arbitrarily fixed distinct integers <[ m: 



1) If, as in 97, the indices are to belong to the GF[p] so that the 

 invariance of O r is numerical and not formal, we must take r <p n in order 

 that our results shall still hold true. Cf. 152. 



