M.-P.-Vol. I.] DICKSON— THE ORTHOGONAL GROUP. 37 



the identity nor N. We may thus suppose it to be not 

 commutative with 7ia, for example. Thus /contains 



S~ X T\% T56 S Z5G 7i2 = S~ l\i S 7i2=Og 7i2 , 



where S§ denotes 1 the substitution 



F 1 =&-a l ia 1 & (*=i--6), 



where $1=0^ — a a . By virtue of (2) we have 



i=l 



By transforming S§ T n by 3456 we may take 8 6 =o. We 

 are thus led to the case treated in the next paragraph. 



13. We have shown that the invariant subgroup /must 



contain a substitution 



5 



S : £'i= 2a ij£j (*— 1 • • • 5)> 

 i=l 



leaving £ 6 » . • • f m fixed. 



By § 5 we may suppose that S is not commutative with 

 every C\ (*=i . . 5). Then if S be not commutative with 

 C\ , for example, and if we suppose m^6, /will contain 



S C\ C§ S C\ Cg = S ~ C\ S C\ = i a C\ , 



not the identity, where T a denotes the substitution 



5 

 T : £',= (?, — 2a il Sa jl f j (3 = 1 . . 5). 



Transforming T C\ by <9 2345 , we may take a 51 = o. If 

 T C\ be commutative with every 7^ Z56 (z',J=i . . 4), 

 then a n = a 2] = a 3 i = a iX , and 4 a 2 n = 1. Thus 7" becomes 



1 Thus 6" replaces £. by ^a.. fy which 7*,, replaces by 

 6 



^^^j + K-^i) (£i-£ a )- 



3=1 



This 5 1 - 1 replaces by 



^ + (- i 2-«n)- v («ji-« j2 )e j . 

 3=1 



