CANONICAL FORM AND CLASSIFICATION OF LINEAR SUBSTITUTIONS. 239 



substitution D with a = /3 is commutative only with the p* n (p n I) 2 

 ns of 6r 3 



ex, y' = ay, z' = by + az + ex (a, I, c,d,em the G-F[p n ~]). 



substitutions of 6r 3 



A substitution D with a =|= /3 is commutative only with the p n (p n I) 2 

 substitutions of 6r 3 



x' = ex, y' = ay, z' = l>y-}-az. 



The total number of substitutions of 6r 3 reducible to the types D 

 is thus 



d) (p-l) (p-l) (p'+l) + (|-1) (p"-2) (jp--l) (r+1)^". 



No two of the jp n 1 substitutions of type E are conjugate under 6r 3 . 

 Each is commutative only with the p* n (p n 1) substitutions of 6r 3 



x' = aa;, 2/' = bx -f a?/, f == ex + &$/ + ^^ 

 The number of substitutions reducible to the canonical forms E is 



e) (^ 



A check on the above enumeration of the substitutions of 6r 3 

 consists is verifying that the sum of the numbers a), b), c), d), e) 

 equals the order N of 6r 3 . 



223. Consider next the group 1 ) 6r 4 of order 



By 221, 6r 4 contains a substitution in whose characteristic deter- 

 minant A (A) = >l 4 a^ cc 2 h 2 a 3 A a 4 the coefficients a 19 a 2 , 3 , 

 4 are arbitrary marks of the 6rF[p ra ], 4 =|=0. According to the 

 possible factorizations of A (A) in the GrF[p n '\ ) we distinguish the 

 cases: I) irreducible; II) linear factor and irreducible cubic; III) two 

 distinct irreducible quadratic factors; IV) equal irreducible quadratic 

 factors; V) irreducible quadratic and two distinct linear factors; 

 VI) irreducible quadratic and two equal linear factors; VII) XI) four 

 linear factors, according to the number of equal factors. Denote by 

 A,, p t marks of the GF[p nf ] not in the GF[p nt ], r < t. For simpli- 

 city, the subscript unity is omitted from the marks a, /3, y, d of the 

 GF[p n ]. The types of canonical forms of the substitutions of 6r 4 

 may be exhibited in the following complete list: 



1) Cf. T. M. Putnam, Amer. Journ. Math., vol. XXTTI, pp. 4148. For the 

 author's treatment of the case n = 3, ibid, pp. 3740. 



