THE COMPOUNDS OF A LINEAR HOMOGENEOUS GROUP. 153 



For example, [] 2 replaces Y 1S by the function 



13 



34 



(Z-Y), 



which becomes Y^ s of the table if we apply the Abelian relation 



= 0. 



13 



12 



13 

 34 



i ii 



<!* 



'32 



C 13 



<33 



Similarly, it replaces Y by the function 



12 

 12 



34 

 12 



T{ 



By means of the Abelian relations 

 = 1- 



34 

 12 



12 

 1 2 



12 



34 



1 2 



1 2 



12 

 13 



34 



34 

 34 



34 

 13 

 34 



12 

 12 



34 

 13 



12 

 13 



Hence Y is replaced by the function Y' given by the above table. 

 It is therefore a substitution on five indices leaving absolutely 

 invariant the function 



= z\ - [i 2 3 4] = r 2 + F 13 r 24 - F M F M . 



For p > 2, the simple group -4(4, p n ) is holoedrically isomorphic with 

 a subgroup of the quinary linear group leaving the quadratic function 4> 

 absolutely invariant. 



This theorem and the results of 163 165 find application in 

 Chapters VII and VIII. 



163. By 155, the quaternary linear group of determinant unity 

 SLH(4:, p n } = G[ is holoedrically or hemiedrically isomorphic with 

 its second compound 6rl,2 according as p = 2 or p > 2. By 

 103 104, 6r{ has as maximal invariant subgroup the group 

 generated by the substitution 



M^: g = pi,- (i = 1, 2, 3, 4), 



where p is a primitive root of ft rf =l, d being the greatest common 

 divisor of 4 and p n 1. The quotient -group LF(4t, p n ) is a simple 

 group of order 



1 

 Cv 



To M/j, there corresponds in 6rl,2 the substitution which multiplies 

 every index by fi 2 and therefore the identity if p = 2 orjp n =4Z + 3; 

 while, for p* = 41 -}- 1 , it is the substitution T multiplying each of 

 the six indices by 1. We may state the theorem: 



