Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

~ 83]

LINEAR GROUP

197

5. Find the general form of a linear transformation in three variables
which is commutative with
(a 0 0
S= 0 a 0, a-Ab.
00 bj
6. Prove that a similarity-transformation is commutative with any
linear transformation in the same variables.
7. Prove that the multipliers of a transformation of finite order are
roots of unity (cf. ~ 116).
82. Remarks. For the purpose of avoiding a possible confusion on
the part of the student of the terms " literal substitution" and " linear
substitution," the term " linear transformation " has here been adopted
throughout.
A different value for cst from that given in ~ 78 is obtained by interpreting the linear transformations A and B in a different manner or by
writing last in the product that transformation which operates first, as
is the custom with functional operators. Thus, Klein, Jordan and Burnside regard the variables in the left-hand members of A, ~ 75, as the
new, and those in the right-hand members as the old variables (the
accents being placed accordingly or entirely absent), and therefore get
Cst= v=a= vsbtv instead of the value given in ~ 78; while Weber, attaching the same meaning to the linear transformations, inverts the order in
the product, writing BA where the authors mentioned write AB. On
the other hand, Frobenius and Schur, though writing ys for x's in the
equations A, ~ 75, interpret linear transformations in the same way and
arrive at the same results as the author of this Part II. The latter has
deviated from his customary notation in papers published on the subject
to the extent of dropping accents of variables that were formerly supplied with them and vice versa.
GROUPS OF LINEAR TRANSFORMATIONS, ~~ 83-87
83. Linear Group. If co is an imaginary cube root of unity,
the six transformations
A==, A As=
0 1 1 j 0 W2J
0 A2 - o J2 0 A (0
A4=, As=, A6=
A 0 0 0 o, 2 0,