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

~ 841

COLLINEATION-GROUP

199

in the latter, the g transformations of G can be arranged into
g/f sets, of f transformations each, furnishing g/f distinct collineations. For, let A1,.., Af, be any set of transformations representing the same collineation, then
A1Al-l=S1, A2Al-l=S2,..., AfAl-l=Sf,
are similarity-transformations and are all distinct. Hence,
if the group F of similarity-transformations S1, S2,..., Sf
contained in G (cf. Ex. 1, ~ 87) is of order f, we have ff'.
On the other hand, if A is an arbitrary transformation in G,
then the f distinct transformations AS1,..., ASf all represent the same collineation, so that f' ~f. Hence f'=f.
The sets of G can therefore be exhibited as follows:
S1, S2,..., 5;
AS1, AS2,..., AS;
BS1, BS2,..., BSf;
To each line will correspond a single collineation.. Moreover,
if the product of a transformation from a set (a) and a transformation from a set (I) fall in the set (y), then the product
of any transformation from (a) and any transformation from
(j) will fall in (7), since the two products merely differ by a
similarity-transformation. Accordingly, the group G is (f, 1)isomorphic with an abstract group H of order h=g/f, namely
the quotient-group G/F (~ 13). Since a collineation in n
variables can be interpreted as a projective transformation in
space of n-1 dimensions by using homogeneous coordinates,
the abstract group H becomes a group of operators of order
h, called the collineation-group corresponding to G.
85. An Example. Take the linear group G of order 8:
1 0 I-1 0 1 I'0 -1
Al=, A2=, A3=,A4=,
0 1 -1 0J 0 -1 1 0
1 ' 0 1 1 0 0 -1
B1 =, B2 =, B3 =, B4 =
0 -1 1 O 0 1 -1 0