Chap. VIII FINITE GROUPS 497 
This reducible curve is invariant for all transformations of 
the group. 
If the order of / be sufficiently high we can get as many 
linearly independent invariant curves of the order of F as we 
want; the linear system of least dimension, given by base points, 
that holds all these will be invariant. We thus get the theorem 
which is fundamental here as the corresponding theorem was 
for continuous groups.* 
Theorem 10] In any finite group of Cremona transformations 
there will always he an infinite linear system of curves given by 
base points which is invariant for all transformations of the group. 
We may follow our previous reasoning step by step and show 
that there must be invariant an infinite linear system either of 
rational or of elliptic curves. If there be an invariant one- 
parameter system of rational curves, these may be carried into 
a pencil of lines, and the transformations are all of the De Jon- 
quieres variety. If all lines through the singular point be 
invariant, we fall back on essentially the problem of the last 
section. If they be permuted, we meet also the problem of finite 
groups of binary projective transformations. These groups are 
thoroughly well known since the classic work of Klein 2 . Such 
groups are simply isomorphic with cyclic groups of rotations 
about a fixed axis, a dihedral group consisting in a cyclic group 
and a reflection in a plane perpendicular to the axis, and the 
groups which carry into themselves the five regular solids. 
If an invariant system of curves depend on two parameters 
it is equivalent to the system of all lines in the plane, and we 
meet the problem of finding finite groups of plane collineations. 
These were first classified by Jordan 2 , the results being com 
pleted by Valentiner 2 . If the invariant system depend on more 
than two parameters, it can be carried into a set of curves of 
order n with a fixed point of multiplicity n— 1 and perhaps 
other fixed points; we are thrown back on a problem of De 
Jonquieres transformations. 
Let us next suppose that we have an invariant system of 
elliptic curves. If these can be carried into quartics with two 
* The most extended researches in the present topic are Kantor 3 and 
Kantor 4 . The works are crammed full of results, but obscurely written and 
not free from errors. A better presentation is that of Wiman. 
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