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

244 THE LINEAR GROUPS IN THREE VARIABLES [C". Xtl

Next we put 0 in place of q-2 of the roots 3, P2,...?
0q3-1, and -1 for the remaining root, thus changing
B'(1+0+2+... +O-1) into B'(1+0+0+... +(-)), so
that this product still remains equal to zero. Similarly, we
put 0 in place of r-2 of the roots y, y2,..., y-1, and -1
for the remaining root, and so on. Proceeding thus, we shall
ultimately change (7) into an equation of the form
A"(l+a+a2+... +aP-l) =0,
where A" contains roots of the form ~-pE only.
Finally, we put 1 in the place of every root a, a2,...,
ap-1, as well as every root Ep. The left-hand member may then
no longer vanish, but will in any event become a multiple of p.
The final value of the expression (8) would be (w+1)(1+1) =2 or 0,
according as co is replaced by 0 or -1.
Notation 1. Any expression N which is a sum of roots of
unity, changed in the manner described above, shall be denoted
by N'p.
3~. We shall now study the effect of these changes upon
the left-hand member of (6). Each of the characteristics
[VS],..., [VSI], being the sum of three (unknown) roots
of unity, will finally become one of the seven numbers 0, i1,
~2, ~3, whereas [V], being the sum of three roots of index
1 or p (cf. 1~), will become 3. The left-hand member of (6)
will thus take the form
(9) [VS]', +3K', +L',p VS]'v +M'v[VS2]'v,
and this number is a multiple of p (by 2~).
The values K'p, L'p, M'p may be obtained by treating them
as indeterminates 0/0. Thus,
ai a2 a3
K= l1 — of2 22 a2 32, K1'p(oal-a 2)(a2-oa3)(a (-a)o Ol
saiA a2A a31I
We find
K'P=-(-1)(-2), L',=(-2), M'= —(y-1),