110 



CHAPTER IE. 



CHAPTER III, 



A GENERALIZATION OF THE ABELIAN LINEAR GROUP. 1 ) 



124. Those linear homogeneous substitutions in the G-F\j) n ] 

 on mq indices , 



87) S: x'ij = ? (& x kl 



t=i 



, . .., f; j 1, ..., 



which, if operating simultaneously upon ^ independent sets of mq 

 variables , the j ih set of which is given the notation 



00 



leave formally invariant the function 



'i'2 ' 



form a group G~(m,q,jp n ), which for q = 2 is the Abelian group 



The conditions upon S for the absolute invariance of O are seen 

 to be those given by formulae 88) and 89), viz., 



88) 



89) 



m 

 V 



i = l 



il 



il 



iq 



/each Is = 1, 2, . . ., g; 

 = ( each j = 1, 2, . . ., m, 

 \ji>J2>- ,jq not all 



125. T/ie inverse of the general substitution 87) o/* 6^(m, g, ^ n ) s 



90) 



, . .., m; s = 1, . . ., 



1) Taken from the author's paper, "A class of linear groups including the 

 Abelian group", Quarterly Journal, July, 1899. The group is mentioned, but 

 not investigated, by Jordan, Traite, p. 219, No. 301. 



