M.-P.— Vol. I.] DICKSON— THE ORTHOGONAL GROUT. 51 



commutative with C\ C 2 ; for we can not have oc\? -\- *u = 1 

 as required by the note to p. 32. Thus / contains 

 S~ l C\ Ci S Ci C\, not the identity and having the 

 coefficient 



xn = — (1 — 2 x n 2 — 2 A'12 2 ) = o (mod. 3), 



We may therefore suppose that x n = o in the substitution 

 S. Transforming by 5 m we may make a second coeffi- 

 cient x\j = o. Then from 



5 



2 aty 2 = 1 (mod. 3) 



j=i 



it follows that four of the x^s are zero. Let x i5 ^ o. 

 Thus S is not commutative with C 5 so that / contains 

 R x C\ C*> not the identity (p. 40). Four of the x 2j 's are all 

 zero or all not zero. Transforming by certain C5 C j5 when 

 necessary, we may suppose that, for example, 



#21 = #22 = #23 , X\i = #12 = #13 = O . 



Hence -R x C\ C5 is commutative with T\i Z13. 

 Hence / contains the substitutions 



( C\ C5 -R x ) i\z in (J?x Ci ^5) T\<i i\a= C\ C5 • Ci C5 = C\ Ci , 



W 1 C\ C 5 WCx C 5 = W , 71a t m = w- 1 c x c 2 wc x c 2 c 3 c*. 



Hence / coincides with H. 



7. Theorem. The Orthogonal Group on 3 indices in 

 the G F \_p n ~\, p^>2, has a sab-group H of order ]/ 2 p n 

 (p 2n — 1) which is simply isomorphic to the group of linear 

 fractional substitutions of determinant unity on one index. 



Let i be a root, real or imaginary, of i 2 = — 1 ; so that 

 i belongs to our G F \_p n ~\ only when — 1 = square. 



Introduce in place of £1, | 2 , £3 the new indices 



Vi = — i li > V2 = h — * h , Vs = h + * h , 

 whence — 7]{ 2 -J- r/ 2 Vs = |j 2 -f £ 2 2 -|- | 3 2 . 



The orthogonal substitution 



3 

 S : f, = 2 * y g, (*' = 1, 2, 3) 

 i=l 



takes the form S\ : 



