LINEAR HOMOGENEOUS GROUP IN THE F[2] etc. 215 



#im =4= 0, transforming by M 2 M m if necessary. Transforming the 

 resulting substitution of the form 



mlfot,... (&i =H, $i m + 0) 



by the substitution .R m , 2, *, we obtain a similar substitution having 

 in 1/1 the additional term xdi m % 2 . 



Recurring to /S", in which d 12 =[= 0, we transform it by T 2J d~ 1 

 and obtain a substitution /S x of K having the form 



Si - | 1; Vi - Aili + -*% + Atl. + % 

 Consider the following product, leaving 1? i^, | 2 fixed, 



"FT= 3,2,^-^3,2,^3 - Qm,2, 



It replaces ?? 2 ^J the function 



in which the coefficient of | 2 equals 11 a~ 1 + /3 12 by 214), since 

 d 12 = 1 and d n = a~ i . Hence W transforms S into the substitution S 2 : 



II = all, <= 0u 61+ ~Si + %+ fti"" 1 ^; - - 



Let f* = ft n cc~ 1 =4= ^- ^ among the substitutions 3,2,^^2,3,1, 

 T^pM^Mz, etc., of Ji, leaving Ij, ^ and ft| 2 + ^ 2 invariant, there 

 exists one, say F, which is not commutative with $ 2 , then K contains 



which leaves ij and ^ fixed. In the contrary case, we find, on 

 equating the functions by which S 2 8 , 2 , p. N^ 3, i and 3j 2 , ^ N^ 3, i ^ 2 

 replace 2 , that 



^3 = (82 + ^23 + f 722)^3 + "23% + ^^23^2' 



By one of the relations 194), we find a 23 =0. Then, if $ 2 be also 

 commutative with T^pMiMs, we must have 3 = | 3 , Vs^^s- 



In proving that K contains a substitution $ =j= I which leaves |j 

 and % fixed, we assumed the existence of the indices 



I,-, ^ (; = 1, 2, 3, m) 



only. But, by the relations 78) and 194) S is a hypoabelian sub- 

 stitution on the indices fj f , ^ (^ = 2, . . ., m). Hence, if m > 4, a 

 repetition of the previous argument shows that K contains a sub- 

 stitution =|=7 involving only the indices |,-, t?i (i = 3, . . ., m). After 

 m 3 such steps, we reach in K a substitution =)= / and affecting 

 only six indices | t -, v\- t (i = m 2, m 1, m). In view of the sim- 

 plicity of the senary hypoabelian groups, K will contain all the sub- 



