216 CHAPTER VIE. 



stitutions of Ji will affect only the last six indices, and, in particular, 

 MiMj, Nij^ (i, j = m %) m 1? m )- Transforming the latter by 

 suitable substitutions P rs (r, s < m, if A = A'), we reach all the 

 generators of Ji. Hence K=Ji, so that J;. is a simple group. 



In view of the importance of the subgroups J" and J^' of the 

 first and second hypoabelian groups respectively, they will be 

 designated by the more explicit notation FH(2m, 2 n } and SH(2m, 2 re ). 

 They are both simple when m 5> 3. The second is simple and the 

 first is composite for m = 2 ( 196198). 



210. MISCELLANEOUS EXERCISES UPON CHAPTERS I VUL 



1. Every m-ary linear homogeneous substitution in the G-F[2] leaves 



invariant the function Sj -j- S 2 + h Sm, where s r denotes the sum of 



the products of the m indices taken r at a time. 



2. An m-ary linear homogeneous substitution in the G-F[p n ] of 

 determinant D multiplies by D the function of the indices 



if 



p n(m 1) p n(m 1) n(m 



51 52 ... 5m 



Hence Y is a relative invariant of the group G-LH(m,p n ). 



3. The structure of the m-ary linear homogeneous group in the 



G-F[2 n ] which leaves |? + i! H f- U absolutely invariant may be 



derived from that of the special linear group SLH(m 1, 2 W ). 



[Take as new indices X = | x -f | 2 H }- and | 2 , J 8 , . . ., &]. 



4. Those substitutions of the hyperorthogonal group G- m ^ n ( 143) 

 whose coefficients all belong to the GF[2 n ] form a group 6r, a subgroup 

 of the group of Ex. 3. Prove that 6r is generated by the binary sub- 

 stitutions ,, . . , . ^ ,_ .f , N . . 



and that G is a solvable group of order 2 7m ( m ~ 1 )/ 2 . 



5. Consider the group (7 of 2 m-ary substitutions in the GF [p n ] , p > 2, 



common to the special Abelian and orthogonal groups. Being Abelian 

 its reciprocal is 'obtained by replacing ,-y, y,-y, ft^, (5,^ by d^-, y^,-, 

 ft'i tyf respectively. Being orthogonal, its reciprocal is obtained by 

 replacing the former by oy$, ft t -, y^,-, ^,- t -. Hence must 



c) ay { = ^-f, ft f = - yji (i, j = l,..., m). 



