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

242 THE LINEAR GROUPS IN THREE VARIABLES [CH. XI1

and a13 =1 imply a =1, whereas S is not the identity. Of the
two cases we shall treat the latter only: the method would be
the same in the former case,* and the result as stated in
Theorem 13 would be the same.
Selecting now from G any transformation V of order p:
ai bi c1
V= a2 b2 C2
a3 b3 c3
we form the products VS, VS2, VS'". Their characteristics
(~ 89) and that of V will be denoted by [VS],..., [V], and
we have
[V] =ai +b2 +C3,
[VS] =aiai +b2a2 +C3ao3,
[VS2] = ala2 + b2a22+C3a32,
[VSI] = alal -+b2a2 +c3a3".
We now eliminate al, b2, C3 from these equations, obtaining
[V] 1 1 1
(5~) [VS] ai a02 a3
[VS2] i12 a22 a32
[VS] aMll ag2 aca'
Expansion and division by (al -a2) (a2 -a3) (a3-al) gives us
(6) [VS'] +K[V] +L[VS] +M[VS2] = 0,
K, L, M being certain polynomials in ai, a2, a3, with the general
term of the type ala2 baa3. Since ai, a2, a3 are powers of a
primitive pth root of unity a (~ 116, 5~), the quantities K,...
are certain sums of powers of a. Moreover, the characteristics
[V]... are each the sum of three roots of unity (~ 81, Ex. 7).
If, therefore, the products in (6) were multiplied out, there would
result an equation of the kind discussed in ~ 116, 6~. The
* The congruence (10) would here be of the first degree in 2.