212 CHAPTER VH1. 



9 4 (nSn j_ 9 2rc 9 ri\ 



2 -(2 +2 -2 J ^gn 



2 W -1 



sets of solutions, and therefore in all 



sets of solutions. But each value of r furnishes two values of (l mm . 



209. Theorem. The hypoabelian groups Ji on 2m > 6 indices 

 are simple. 



Let K be a self- conjugate subgroup of Ji containing a substitution 



.7=1 .7=1 



not the identity J. We first prove that .ZT contains a substitution =f= 

 multiplies ^ % a constant. Let $ replace ^ by 



where by 194), 



m 



213) ?<xi J ru+*>lm+ *-ylm= o. 



If /i =f= a nli; we have one of the following three sub -cases. 

 a) y 11 =|= 0. Then J^ contains the product 



T = TI ? yu - 1 1? 2 , i, au Q 2 , 1, y w JR/n, 1, ^ m w, 1, n m 



which replaces ^ by y^ 1 ^ and ^ x by the function 



H 



This equals f l} since the coefficient of | x is congruent to cfj! modulo 2, 

 in virtue of 213). Hence K contains S ^T l 8T f which replaces 



61 b y yn 1 ^- 



If J^ contains a substitution T x which leaves ^ and ifj 1 fixed 

 and is not commutative with 8 19 K will contain the product 



which leaves ^ fixed. Suppose on the contrary that S is commu- 

 tative with every substitution of Ji which leaves | x and ^ fixed. 

 Among the latter are E^^ y . and 3, 2, x- If we equate the two 

 expressions by which SiE^s^. and JR2,s,xi replace t; 3 , we find 



Similarly , if S 1 be commutative with Q^ 2 , x; we have 



