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

340 INFLEXION POINTS OF A CUBIC CURVE [CH. XVIII

three collinear inflexion points are in the enlarged domain;
the same is true for their products by threes giving the inflexion
triangles h+24rf, so that each root r of (6) is in that domain.
It now follows from the theorem of Jordan (~ 167) that the
adjunction to R of the four roots of (6) reduces the group G
of X for R to an invariant subgroup Z of index 24 under G,
such that G/C is simply isomorphic with G24.
This group S was shown to contain a transformation (15)
of period 3, necessarily a translation (13). By interchanging
t and 77 if necessary, we may assume that ~ is altered. Then
the translation or its square is of the form
i,-~+1, v'-+t.
Introducing ~ and -t~ as new variables, we obtain a group
1i conjugate with 2 under L and containing the translation
'=- +1, ='-rl. The only transformations (12) which are
commutative with this one are those with a=1, A=0, B 0,
b, c, C arbitrary, 2.33 in number. The above translation is
transformed into its inverse by i' —, H'=N. Hence exactly
4.33 transformations of L transform into itself the cyclic group
of order 3 generated by it. Since this number is one-fourth
of the order of L, a subgroup of index 3 under L cannot transform this cyclic group into itself.
But Z is of order 6 or 18. In the first case, S contains
a single cyclic group of order 3, which is therefore invariant
under G; while G is of order 24.6 and hence of index 3 under L.
Thus the first case is excluded by the preceding result. Hence
2 is of order 18 and G=L.
THEOREM.* If the coefficients of a cubic curve f=0 are
independent variables, the group of the equation upon which
depends the nine points of inflexion [r1l], i, 7 =0, 1, 2, for the
domain of the coefficients, is the group of all linear transformations
on ~ and r7 modulo 3.
After the adjunction of the roots of the resolvent quartic
(6), the group is that of the 18 transformations (15). The
* Stated, but not completely proved, by Weber, Algebra, ed. 2, vol. 2, pp.
416-7. The proof is due to Dickson, Annals of Math., ser. 2, vol. 16 (1914), pp.
50-66.