CHAPTER L 



GENERAL LINEAR HOMOGENEOUS GROUP. 1 ) 

 97. First definition. Consider the p nm letters, or symbols, 



fc*i-r>, 



characterized by m indices, each running through the series of marks 

 of the G-F[p n ]. The general linear homogeneous substitution A on 

 the m indices |,- with coefficients in the GrF [p n ] replaces the letter 

 Ify # . . ., z m by l$' v f '#..., s' m where 



m 



A: 8-.V*i (-l,..,,m) 



the coefficients ,-> being marks of the field. But A will indeed 

 permute the p nm letters if, and only if, the determinant of A is not zero, 



\A =K, 4=0. 



In fact, there must be one and only one system of m indices which A 

 replaces by a given system J' and hence an unique set of values J^ 

 satisfying the equations 



Let B denote a second substitution with coefficients in the GF[p u \ 

 B: 



where 



1) Jordan, Trait des substitutions, Nos. 119, 169; author's dissertation, 

 Part II. Cf. 95 above. 



