232 CHAPTER X. 



Similar remarks hold for the indices g,-^ and ifj/y. We may now state 

 our results in the following form: 



Theorem. To determine tlie most general linear homogeneous 

 substitution T on m indices with coefficients in the GF[p n ~\ which shall 

 be commutative with a particular one S, we apply the transformation 

 of indices which reduces S to its canonical form S 1 and T to some 

 form T v Then S 1 may be expressed as a product 



S = 7? ^l . . . 1?* ! go l &-1 #0 #1 . . . #9 1 



where each substitution ??/, /, #,- is defined thus: 



rii\ tf ia = KiVjia, rfiA = K^A + -K^ 4-1 (for every a, A) 

 /: gi6=jLfgt-&, tiBL&B + I'&Bi (for every b, B) 

 #: il>ic= Qifac, ^ic=Qi^ic+Qi^ic-i (for every c, C). 

 The most general T v must be expressible as a product 



TI = H H x . . . H A _i Z Z-t . . . Z;_i YO ^ . . . M / ? _ 1; 

 e individual substitutions have the forms: 



Z,: a-=><> 0' -!-. ft 



0=1 



^e coefficients d ju , Q JV , <5 jm being polynomials in K , L^ Q Q) respectively, 

 with coefficients in the GF[p n ~]. Furthermore, H must be commutative 

 with ?/ , Z with g , VQ w^ # . 



Inversely, if these conditions on H^ Z z - ; Vf &e satisfied, then the 

 substitution T corresponding to the product T will be commutative 

 with S and will have its coefficients in the GF[p n ~\. 



218. In order that the substitutions H and T/ O be commutative, 

 it is necessary and sufficient that, for every a, A and A', 



234) d aA = 0, d^ la _! = 0(a>l), ^_ la = 0, 8 A -iA'-i-8 AA : 



Indeed, ^ H and H O T/ O replace ^ by the same function only if 

 every 3 aA =0. In order that they shall replace ^ A by the same 

 function, we must have 



If u is not of the form A' 1, it must be of the form a 1 or else a. 



