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

~ 1761

GROUP FOR THE INFLEXION POINTS

335

rows, columns, and positive and negative terms of the expansion
of the determinant
1 4 7
147
258.
369
Henceforth we shall denote the abscissas of the nine points
of inflexion by the nine symbols [ 77], where =0, 1, 2 and
7=0, 1, 2. Then the abscissas of collinear points of inflexion
are those in the rows, columns, and positive and negative
terms of
[00] [01] [02]
(10) [10] [11] [12]
[20] [21] [22]
and have the sum of their first indices divisible by 3, and also
the sum of their second indices divisible by 3.
Hence G is a subgroup of the group L of those substitutions
on the nine roots which replace any three distinct roots [il],
i=l, 2, 3, for which
(11) 1+~2+ 3-=0, 1+1 2+ 3 —O 0 (mod 3),
by three distinct roots [i's 7'7] also satisfying congruences (11).
We obtain a substitution of L if we take
a b
(12) i'-ai+br7+c, q'-A-+B7-+C, A B 0 (mod3),
A B ~0 (mod3),
where a,..., C are integers. For,
3 3 3 3
1i"=aa ~ +b^ +3c-O, 0-O (mod 3).
i=1 {=1 i=l {=1
Further, the [~' I'"] are distinct. For, if 2'i —2, 7/1 7 /2,
then
a(l - 2) +b( l-772) -0, A (1 - 2) +B(q- -2) -0 (mod 3).
But the determinant aB-bA is not congruent to zero. Hence
— ~2, 717 -2, contrary to hypothesis.