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

336 INFLEXION POINTS OF A CUBIC CURVE [CH. XVIII

Conversely, every substitution S of L is induced by a linear
transformation (12) on the indices. Let S replace [00] by
[c C]; the same is true of the substitution P induced by
(13) i'-=+c, '-?,+C (mod 3).
Hence S=TP, where T is a substitution of L which leaves
[00] unaltered. Let T replace [10] by [aA], where therefore
a and A are not both zero. Thus we can find two integers
b and B such that aB-bA is not divisible by 3; if a 0, we
may take B=1, b=0; if A 0, we may take b=1, B=0. Then
the substitution P' induced by
(14) i'-aE +b-, r' —A+Br,, aB-bAO0 (mod 3)
replaces [10] by [aA]. Hence T=T'P', where T' is a substitution of L which alters neither [00] nor [10] and hence not
[20], in view of the first column of (10). Let T' replace [01]
by[de], so that e#O. The substitution Pi induced by
=' +dr, l' - e7
leaves unaltered each [~0] and replaces [01] by [de]. Hence
T'P1-1 leaves unaltered each [0] and [01] and is the identity.
For, it leaves fixed [02] by the first row of (10), and hence [21]
and [12] by positive terms of the expansion of determinant
(10), and then [11] and [22] by the second and third rows. Hence
T'= P and S=P1P'P, so that S is induced by a linear transformation (12).
Now [cC] was any one of 3X3 roots, [aA] any one of 32-1
roots, and [de] any one of 3X2 roots.
THEOREM.* The group G of the equation X for the abscissas
[77] of the nine points of inflexion is a subgroup of the group
L of all of the 9X8X6 linear transformations (12) on i, r.
The 32-1 incongruent linear homogeneous functions of t
and nr with integral coefficients modulo 3 are
+t, ~=~, +=(~+r), i( -7).
* Jordan, Traite des Substitutions, p. 302, where L is defined to be the group
leaving (formally) unaltered the cubic function given by the sum of the products
of the roots in each row, etc., of (10). But formal invariance may well introduce some confusion since the roots are not independent. For a wholly different
determination of L, see Weber's Algebra, 2d ed., vol. 2, p. 413.