CANONICAL FORM AND CLASSIFICATION OF LINEAR SUBSTITUTIONS. 241 



Table giving the form and number C of the substitutions of the 

 group 6r 4 commutative with the various types of canonical forms: 



I 



II 

 III 



vn 



HI, 



ra, 



IX, 



ix s 

 K 3 



X, 



XI, 



XI a 



QX 

 QX 



ax 



ax 

 ax+by 



ax 



ax 



ax 



ax 



ax 



ax 



ax 



ax 

 ax-\-ez 



ax 



ax 



ax 



ax-\-by 



by 



cx-\-dy 

 by 

 by 

 by 

 by 

 by 



bx-\-ay 



l)x-\-ay-\-fz-}-ew 

 bx-\- ay -\-fz-\-ew 



bx + ay 

 bx-{-ay 

 gx-\-hy 



GZ 



P 



Q P W 



QZ 

 QZ 



QZ 



CZ 



cz 



cz-\-dw 

 cy + bz 



cx-{-by-\- az-\-ew 



gx+cz + l 

 arbitrary 



cz 



cz-\-dw 

 cz-\-dw 



aw 



dw 



dz-\-cw 



ez + fw 



dy-{-cz--bw 



fy + dw 



dx -\-cy-\-bz-\- aw 



fx-\-dw 



hx+gy+dz+cw 



hx-\-dz-\-lw 



ez-\-fw 

 ez-\-fw 



i 1) (p2n 2? n ) (p n - 



N 



Here ;, belongs to the GF[p^ n ]^ ^ to the G-F[p* n ], Q, <5, x, x to 

 the GF[p* n ], and a, b, c, . . ., j belong to the G-F[p n ~]. If M denote 

 the number of distinct canonical forms in a general type, and C the 

 number of substitutions of 6r 4 commutative with each, the number 

 of substitutions of 6r 4 reducible to that type is MN/C. The sum 

 of these numbers is found to equal N, the total number of the sub- 

 stitutions of 6r 4 . 



DlCKSON, Linear Groups. 



16 



