CHAPTER X. CANONICAL FORM AND CLASSIFICATION etc. 221 



213. The preceding methods may be employed 1 ) to investigate 

 the group G q of linear substitutions S on rq indices with coefficients 

 in an arbitrary field which leave absolutely invariant the function 



For q > 2, it is seen that S = A B, where A merely multiplies each 

 index |/^ by a constant, while S is a permutation on the indices I,, 

 having the imprimitive systems 2 ) 



232) n , | 12 , . . ., l9 ; | 21 , 22 , . . ., J-22J ; 8rl> r2, . ., %rq- 



The substitutions A form a commutative group which is transformed 

 into itself by every substitution S and is therefore self - conjugate 

 under G q . The quotient -group is the group of the substitutions B. 

 The latter has a self-conjugate subgroup R formed by the direct product 

 of r symmetric groups, the general one being on the q letters | fl , 

 !**>?&) ^ ne quotient -group {L}/R is a symmetric group on 

 r letters, viz., the r sets 232). The structure of the group G q , q>2, 

 is therefore completely determined. The result is essentially different 

 from that for the case q = 2 (see 195). 



CHAPTEK X. 



CANONICAL FORM AND CLASSIFICATION OF LINEAR 

 SUBSTITUTIONS. 



Canonical form of linear homogeneous substitutions*), 214 216. 

 214. Consider a substitution with coefficients in the GrF[p n ], 



S: tt 



In order that S shall multiply by a constant K the linear function 



we must have 



or 



Kfy (j = 1, . . ., m). 





1) Proceed. Land. Math. Soc., vol. 30, pp. 200 208. On pp. 203 204 the 

 numerical factors are incorrect; C should equal tj_l t t \ . . . ttl The proof 

 however is valid. 



2) S replaces the indices of any set &i, |2, . . ., &? by indices all in one set. 



3) For n = l, the results are due to Jordan, Traite, pp. 114 126. The 

 simple proof by induction of the fundamental theorem is due to the author, 

 American Journal, vol. 22, pp. 121137. 



