LINEAR HOMOGENEOUS GROUP IN THE JF[2] etc. 205 



The substitutions 206) are evidently powers of 207), p being a 



primitive root of # 2 "+ 1 = 1. Hence the substitutions 202) are powers of L. 



Inversely, every substitution 206) for which T 2 "+ 1 = 1 may be 



transformed by 205) into a substitution 202) of the GF[2 n ]. In fact, 



a + d = r + T- 1 , a = x + (x -f tr" 1 )*, d = T -i -f ( r + r -i)<y, 

 so that a -+- d belongs to the 6r.F[2 n ] and likewise a since 



The number of substitutions 206) is 2 n -f 1. The number of sub- 

 stitutions 202) is therefore 2 W +1. Furthermore M m (| m ^ m ) takes 

 the form 



We have therefore a new proof of the results at the end of 202. 



It is worth while to verify independently that the number of 



substitutions 202) is 2* + 1 according as A = ti or A = 0. We have 



only to determine the number of sets of solutions in the GF\2 n ] of 



208) cc$ + tfa*+H 2 d*=L 



The result for the case A = being evident, we suppose that K = A'. 

 The left member of 208) vanishes only when a = d = 0; for, otherwise, 



would be reducible in the field, contrary to the irreducibility of 203). 

 Hence each of the 2 2n 1 sets of marks a 1? S 19 not both zero, in 

 the GF[2 n ] will make 



Then will ajn, ^i/ 3 ^ be a set of solutions of 208), and inversely 

 every set of solutions of 208) may be so obtained. Hence, if ^ = A', 

 the number of distinct sets of solutions is (2 2 n I)/ (2" 1). 



204. We can now readily determine the order Q^J n of ft. The 

 number of distinct linear functions /J by which the substitutions 

 of ft can replace (^ is P$ n 1, if Pi, denotes the number of sets 

 of solutions in the G-F[2 n ~\ of 197). For m > 1, the pair of equations 



has (2"+ 1 l)P^l.i )n sets of solutions when t = and has 



sets of solutions when r runs through the series of marks =)= of 

 the GF[2 n ]. We have therefore the recursion formula (m > 1) 



P^n= 2-Pi a li |l , + (2- - 



