184 CHAPTER VII. 



Reversing relations 165), we find 



167) r u =s,-.rj,, r 34 =-i 1 



Written in the new indices, the substitution Q' becomes 



The coefficients 0, 7p, p/7 belong to the 6r.F[^ w ] since 



Hence Q" belongs to the GF[p*] and has determinant unity. 



If, in 162, we set Y= J^, Z^=. 7| 6 , we obtain the present 

 transformation of indices 165). Hence, if we express any substitu- 

 tion []. 2 of A^* in terms of the new indices, we obtain a substitu- 

 tion, not involving | 6 , the matrix of whose coefficients is given in 

 162. Hence A^ 2 is transformed into a group A! 1 of substitutions 

 belonging to the GF[p n ~] which do not involve | 6 and which leave 

 absolutely invariant , 2 ^ 



fel ~t~ -Ms-^24 -M4-^23' 



In order that A" shall contain the substitution 



A-. 



it is necessary and sufficient (by 164) that o^ n + 1 be a square in 

 the 6rP[^) ?l ] and hence that o be a square in the 6r_F[p 2n ], 



C) (p n +i)(p n -i)/2 = -J- 1. 



Hence the group 6r", given by the extension of A" by Q", will contain 



if and only if o be a square in the G-F[p 2n ~\. Now ." leaves 

 - || -|- J 2 || invariant and is therefore an 16 . We proceed to prove 

 that, if a be a square in the G-F[p 2 *], every K is a Q"l% and every 

 Qif & is a K, where a, ft belong to the GrF[p n ~]. Let, in fact, 



168) ^- 



Since a)^ 2 "- 1 )/ 2 ^ 1, we see that a and /3 belong to the GF[p n ~]. Also 



169) 



170) 

 Hence has the form $?,'{?, where a, /? are defined by 168). 



