170 CHAPTER VII. 



Let OfJ be a particular substitution which extends Q ti to 0^-, 

 the values Q, (5 depending on A. Consider the subgroup O^(W,JP TC ) 

 of On(m,p n ) generated by the substitutions 



CfA Of/ Of* ft A M - 1, , ; + ;, * + i) 

 where a, /3 take all the values in the G-F[p n ~\ satisfying 2 + -7-j8 2 =l, 

 the generator JR, W or V being added in the exceptional cases of 173. 



182. Theorem. - - The order of Ol(m,p n ) is at least half of the 

 order of O ft (m,p' l ). 



By the theorem of 173, every substitution of O^m, p n ) has 

 the form 8 ^0^0},^, . . . 



where the hi (and the h\, h lr , h, . . . below) are derived from the 

 generators of 0;(w, p n ). For m > 2, 0^{ and 0%7* = 0l\{ are 

 reciprocal. Hence 



/-vO, O iT\Qi O /^j 6 /"ki O 7 /~kO G 



ij = Off Oil i il 2 = \ Oil 2 - 

 Hence 



Furthermore, Of; \ is commutative with every Q% and every C?/> 

 * and j > 2. Since the square of Oi ? ' is $fj*> whose reciprocal is 



Aside from the above exceptional cases, we may conclude that 8 is 

 of the form h or else h - 0J. We treat next the exceptional cases. 

 1. For^ w =5 ? m > 3, f& = l, the additional generator is JR 123 , 

 and the only Qfcf are (ftp 1 ==dCj and ^f/'=J. Since 



00, 1 m /Tf 

 ,; -M/^ij 



where T,-^- = (|-iy), is not in ^, it may be taken for O^'/. To complete 

 the proof that 8 = h or hO% J , we note that 



T 12 C 1 - li 12S = Uj C/3 Ji 123 jT 23 (7 2 T 12 C l ' C l C 3 - T 12 Cj . 



2. For p n =3, m^> &, p = 1, the additional generator is Tf" 1234 . 

 The remarks of 1 apply here, if we replace the last formula by 



T 12 (7j W 12U = C C 2 W^ 2B4: T^Ci . 



3. For ^ w =3, m>3 7 p = v = 1, the additional generator is 

 F 12m and the only 0? (a 2 - ^ 2 =1) are C t C m and J, the only $$ 

 being 1. We may take OfJ = T {j d (i, j < m) and 0?'^ = dC m . To 

 complete our proof, we use the formulae 



