88 CHAPTER I. GENERAL LINEAR HOMOGENEOUS GROUP. 



determinant unity or a particular not -square v if p > 2. In fact, 

 if A is a square, ji 2 A may be made equal to unity by choice of ^ 

 in the field; while for A a not- square , |tt 2 A may be made equal to v. 

 If p n > 3, the group LF(2,p n ) of all linear fractional substitutions 

 in the GF\p n ] of determinant unity (when in their normal forms) is a 

 simple 1 ) group of order 



P^^ / 2 l according as p > 2; p = 2). 



There are p n (p 2n 1) linear fractional substitutions of determinant =j= 0. 



From the formula of composition of binary linear homogeneous 



substitutions ( 97) 7 we derive the product SS of linear fractional 



substitutions S = | -? 



= 



" 



Hence if /S> operate first and S L afterwards , the product SS 1 is 2 ) 



109. The quotient -group f/T may be readily represented as a 

 permutation -group on q =::- (p nm 1) -;- (p n 1) letters 3 ). Of the 

 p* 1 letters l^ ^ ...,s m in which % lf | 2; . . ., ^ m denote marks of 

 the GF[p n ] not all zero, we combine into a single system the 

 p n l letters l^ ^ . .. ? ^ m in which p runs through the series of 

 marks =(= while ^, | 2; -, Im denotes a set of fixed marks not all 

 zero. Any linear homogeneous substitution on |i ; . . . ; % m with co- 

 efficients in the field replaces the letters of any one system by letters 

 all of some one system and therefore permutes the q systems amongst 

 themselves. In particular ? the substitutions M^ do not displace any 

 system. Hence the group f of substitutions of determinant unity 

 corresponds to a permutation -group on the q systems , which represents 

 concretely the quotient -group 



1) Cf. Moore, Congress Mathematical Papers, pp. 208 242, Bull. Amer. 

 Math. Soc., Dec. 1893; Burnside, Proc. Lond. Matb. Soc., vol. 25, pp. 113139 

 (Feb., 1894); also see 261 below. 



2) For the same product of matrices , the notation S t S is sometimes used, 

 S operating first. 



3) Compare the method .of 228, 224; also, for m = 2, that of 239. 



