GENERAL LINEAR HOMOGENEOUS GROUP. 87 



108. From the linear homogeneous substitution A of 98 on the 

 arbitrary variables 1, 2; -, % m , we obtain the linear fractional substitution 



-{- Ui m lXml-{- Kim ,. ., ., N 



j - - i - ( I = 1 , . . ., m 1 ) 



(- a mm i X m i -f a mm > 



upon setting #,-= |//| m for i = 1, . . ., w 1. It being only a question 

 of the ratios of the coefficients a^ in A', its determinant a,-; is 

 determined only up to a factor (JL, (JL being a mark =j= 0. Also, ^4.' 

 is the identity if, and only if, A be one of the p n 1 substitutions 



The products -M^J. and no other linear homogeneous substitutions 

 correspond to the same linear fractional substitution A'. Hence the 

 group G GLH(m,p n ) has (p n 1, 1) isomorphism with the group L 

 of the substitutions A. If Q denote the order of 6r, the order of L 

 is Q -f- (p w 1). To the subgroup f formed of the substitutions of G 

 having determinant unity there corresponds a subgroup A of L com- 

 posed of those of its substitutions whose determinant is an m th power 

 in the field. If d be the greatest common divisor of m and p n 1, 

 there are exactly ^substitutions of the form 'M.^ in f and they form 

 the group H ( 103). Hence f has (d, 1) isomorphism with A. The 

 order of A is therefore Q -]- d(p n 1). Aside from the cases (m^p^ = (2, 2) 

 and (2, 3), H was shown to be the maximal self -conjugate subgroup 

 of f; hence A has no self - conjugate subgroup other than itself and 

 the identity and is therefore simple. 



The group LF(m, p n ) of all linear fractional substitutions in 

 the GF [p n ] on m 1 variables and having determinant unity or some 

 m th power in the field has the order 



d being the greatest common divisor of m and p n 1. It is a simple 

 group except in the two cases (m, p n ) = (2, 2) and (2, 3). The group 

 of all linear fractional substitutions of determinants not zero has d times 

 the order of LF(m 7 p n ). 



The notation LF(m,p n ) emphasizes the point that the essential 

 quality of the linear fractional substitution lies in the matrix (a/y) 

 of degree m and not in the m 1 variables x\, . . ., x m i which play 

 the ro le of indeterminates. For m = 2, we use the suggestive notation 



In virtue of the identity of the two substitutions 



C" any mark + 0) 

 * 



y, 



of determinants A and ft 2 A, we may choose ft so that the substitution 

 takes its normal form, viz., of determinant unity if p = 2, but of 



