CHAPTER II. THE ABELIAN LINEAR GROUP. 



89 



CHAPTER H, 



THE ABELIAN LINEAR GROUP. 1 ) 



110. A linear homogeneous substitution on 2m indices with coeffi- 

 cients belonging to the GF\j) n ^ is called Abelian if, when operating 

 simultaneously upon two sets of 2m indices , 



fc . fc / __ -r c> ..,\ 



it leaves formally invariant up to a factor (belonging to the field) 

 the bilinear function 



74) <p 



The totality of such substitutions constitutes a group called the 

 general Abelian linear group 2 ) GA(2m, p n \ These of its substitutions 

 which leave cp absolutely invariant form the special Abelian linear group 

 SA(2m, p*). For other definitions of these groups see 160 below 

 and the author's article , Transactions of the American Mathematical 

 Society, vol. 1, pp. 3038. 



The conditions that the linear substitution 



75) 



S: 



ii 



(i=l, 2, ..., w) 



shall leave 9? formally 3 ) invariant up to the factor p are 



76) ' 



KJJ Ctijt 



Pij Pik 



0', * 



0. 



For w = 1, the Abelian group G t ^.(2, jjf*) is evidently identical 

 with the general binary linear homogeneous group GLH(2, p n \ In 



1) Investigated by Jordan, Traite', pp.171 186, for the case n=l; by 

 the author, 'Quar. Jour, of Math., 1897, pp. 169 178, ibid., 1899, pp. 383 4, for 

 general n. 



2) To distinguish these groups from the ordinary Abelian, i. e. commuta- 

 tive, groups, we prefix the adjective linear. The Abelian linear group is not 

 commutative in general. 



3) The indices & and rji are treated as arbitrary quantities. Formal in- 

 variance is used in antithesis to numerical invariance. 



