M.-P.— Vol. I.] DICKSON— THE ORTHOGONAL GROUP. 3 1 



a, fi . f f, = (a 2 -/3 2 ) £-2 a/3£ 2 



I, 2 ' lf 2 = 2a /3£ 1 - r -(a 2 -/3 2 )f 2 . 



Thus, Q 7 ' 8 Q a ' ^ = Q a ^ S, a o-L-/3 7 

 1,2 1,2 1,2 



Since Q a,/3 '= Q a >& if and only if a = ± a', £ = ± £', 



1,2 x > 2 



the order of the group Q i2 is y 2 (p Q — e). 



Let C K denote the orthogonal substitution affecting only 

 £i, whose sign it changes. Then Ci C 2 belongs to the 

 group Q12. Let 71 2 denote the transposition (fi £ 2 ). Then 

 7i 2 C\ belongs to Q12 if and only if 2 is a square, so that we 

 may have a 2 = ft 2 — yi . 



For p n =8l± 3, 2 = not-square, T\ 2 d serves to extend 

 the group Qi >2 to the group O i2 . Since T x % C\ transforms 



Q a,/? intoQ"'^, Q a '^ intoQ a ' _/3 , for ;>2, it is com- 

 i,i 2, j 2, j i,J 



mutative with the group Q of all the Q u . Hence the group 

 H resulting from extending Q by the alternating group on 

 the m indices is a sub-group of index 2 under the group G. 

 For^ n =n, 13, 19, 29, I find (see appendix) that every 

 7jj T kl is already contained in Q; while for p a = 3 or 5 this 

 is not the case, the Q Uj being merely products of an even 

 number of the C k . 



4. Theorem. For m~><\, p n ~>$, py-2, the group H is 

 simple if m be odd and has the factors of composition 2 and 

 one-half its order if m be even. 



For m even, H contains the invariant sub-group of order 

 2 generated by JV, viz. : 



JV : t\=—^i (i=i,2,..m). 



Suppose H has an invariant sub-group I containing a sub- 

 stitution S ', not the identity and (if m be even) not JV. It 

 will be proved that I coincides with H. 



5. Suppose first that ,5", given by formulae (1) and (2), 

 is commutative with every product C x C\. Then, for^>2, 



«ij = o, a\=i (i,J=i, . . m,i^:j). 



