CANONICAL FORM AND CLASSIFICATION OF LINEAR SUBSTITUTIONS. 235 



* = K + a 2 + <*3 H ----- h <*r + l) + (2 2 + 8 H ----- H r+l) 



- - -f 



the # th parenthesis giving the number of such d {j in the # th row of 

 rectangles. On account of the equalities among the a's, we find 



CD = ^ A| + 4, A 2 (A, + 2AO + 4,A 3 (A 8 + 2 A x + 2A 2 ) + - 



Excluding also the A* + A| H ----- h A? coefficients in the determinants 

 Di o , there remains the following number of wholly arbitrary d {j : 



Q =ti(Aa- 1) + 2^*! + 24 ^3(^2 + AO + 



Each one of these Q coefficients may take p nk values. The total 

 number of substitutions H is therefore 



f(a lt . . ., a r+1 , ft, j) = Q (A u p) Q (A,, JB-*) . . . Q (i,, ^*) . p*. 



T/^6 ^a? number ofm-ary linear homogeneous substitutions T in the 

 GF[p n ] commutative with a particular one S, whose canonical form is 

 expressed in the notations of 217, is given ~by the product 1 ) 



f(a ly . . ., o r+1 , JG, p n ) f(b l9 . . ., 6 -+ i, I, p n ) - f(c 1} . . ., (% +1 , g, ^ w ) . . . 

 Recurring to the above example, ^ =3, a 2 = 3, 3 = 2, we have 

 f (a,, a,, a s , Jc, p") = (p***- 1) (p* - ^) , (p* - 1) - ^*, 

 as is directly evident from the form of H and its determinant. 



220. As an important example, suppose that S has the canonical 



form 



The most general substitution Tj commutative with /S replaces ?? , 

 Jb>--v^o b y ^(^0)^0? ^(-^oKo^-v (>(Co)^o respectively, in which 

 the coefficients of the functions x, A, . . ., p belong to the 6r.F [jp w ]. 

 If jK", i, . . ., Q be primitive roots of the Galois fields of orders p nk , 

 p nl , . . ., p n * respectively, we may set 



1) This result is in accord with that of Jordan, who treats the case n = l. 

 His method of proof is merely illustrated by the consideration of a particular 

 example, Traitd, pp. 128 136. Moreover, it does not give the explicit form of 

 the commutative substitutions. 



