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

~ 87]

LINEAR FRACTIONAL GROUP

201

It is easy to verify that the set of transformations in the last
line form a linear group (G2) which is (n, 1)-isomorphic with
the collineation-group G; in fact, if AB=C, and a, 3 arbitrary
accents, then
A 2(a)B2() =C2(1,
where y is fully determined.
As an illustration, let G be the collineation-group
0 -2
A=(1, 1), B=
2 0
We find
A (1) (1) A 1(2) (-1, -1); B( )=( ( 2, 2) B1(2)=(-i, -2)
and therefore
G2: Am=(1, 1), A2()=(-, -1);
r0 -1 0 I'
B21()=, B2(2)= F.
1 0 1 o.
It may, however, be possible to find a subgroup of G2 of
lower order whose corresponding collineation-group is likewise
G, and whose transformations also have unity for the value
of their determinants (cf. ~ 110). This will be the case if G
is a group of odd order in two variables (~ 97). When in the
future we mention a group of linear transformations of determinant unity corresponding to a given collineation-group or
linear group, we shall mean such a group G3 of lowest possible
order, having the same collineation-group as that given or as
that corresponding to the given linear group.
87. Linear Fractional Group. When only the mutual
ratios of the elements of the matrices are of importance and not
their actual values, we may adopt another mode of representing
the operators, namely by writing them in linear fractional
form. Let a given linear transformation be

A:' x,=acsxit+... +asnx' (s=1, 2,.:..,),