CANONICAL FORM AND CLASSIFICATION OF LINEAR SUBSTITUTIONS. 231 

 By a similar argument, the first and second equations give 



Equating the coefficients of the ^'s in the above identity, we 

 find analogously that every F^ = Q. Hence T^ replaces rj ia by 



a a > W (X = 1, i + 1, a, + a 2 + 1, . . .). 



Consider any a such that a + 1 is an A and equate the functions 

 by which T l S l and Sj^T^ replace ^a+i- Among the relations occur 



- JD^ 1 ! = X, Dft- 



zi^i; +1 -ii.E: +1 



ip- TTna-fl r T7t* + l i T 771*0+1 

 ,/Lj JZ/J _i = JLa JEtf J5_i -f* JL/^ JZ/^ ^ 



From these three pairs of equations we find (as above) respectively 



Hence T A replaces iy,- +i by a function of the ^ tt only. 



Considering any a such that a + 1 and # + 2 are of the set A, 

 we find by the same method that T replaces ^< a +a ^y a function 

 of the rii u only. We readily verify that, if T x replaces ^f a +j by a 

 function of the ^- M only, the same will hold for ^ a _j_d_|_i. Since the 

 series a, a + 1 ? a + 2, a + 3, ... yields every integer, we have proven 

 that T! must replace each ^ by a function of the ^/ w only, if jPj 

 shall be commutative with >S 1 . 



Similarly, T must replace each &j by a function of the g/ r only 

 and each ^ by a function of the il> iw only. 



When we return from the indices ^^-, ^, ^ to the initial 

 indices | 1; . . ., | m , ^ becomes, by hypothesis, a substitution $ having 

 its coefficients in the G-F[p n ~]. Under what conditions will T, T t in 

 the indices |,, have its coefficients in the GF[p n ]? We have shown 



that 2\ replaces % by a function of the form ^ Dl^^u- Recurring 



to the properties 1) and 2), 214, of the indices 17^, we must have 

 as the D l / u certain polynomials in the quantity K t with coefficients 

 in the GF[p n ], such that 



