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

224 THE LINEAR GROUPS IN TWO VARIABLES [CH. X

INVARIANTS OF THE LINEAR GROUPS IN Two VARIABLES,
~~ 104-105
104. General Theory. A homogeneous function of the variables xi, X2 of a group
G: S1, S2,..., Sg,
is called an invariant of G (or we say that G leaves f invariant)
whenf is transformed into a constant multiple of itself by G:
(f)Sj = cf.
Let f be resolved into linear factors. These are permuted
among themselves by G, and the product of a set of them which
are permuted transitively will evidently furnish an invariant
by itself. This invariant, say F=flf2... fh, can readily
be constructed by operating upon one of the factors fl by the
transformations of G, and we shall call it a fundamental invariant. Any invariant is accordingly a product of fundamental
invariants.
If fl be selected at random, the corresponding fundamental
invariant is evidently of degree g. To obtain fundamental
invariants of lower degree we make use of a theorem of transitive substitution groups, namely that the ratio g~/h is the order
of that subgroup of G which leaves fi invariant.
Now this subgroup, G1, must be abelian. For we may
change the variables, introducing fi as one of the new variables,
say xi. Then G1 must appear in the form of a reducible group:
a 0
b c
and can accordingly be written as an intransitive group
a 0
0 c
But this is the canonical form of an abelian group. It follows
furthermore that two subgroups, Gi and G2, having in common
a linear invariantfl, generate an abelian group.