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

198

GROUPS OF LINEAR TRANSFORMATIONS [CH. IX

form a group of order 6 which is isomorphic with the symmetric group in 3 letters (cf. ~ 1). Here A1 is the identical
transformation; the inverse of A3 is A5, while A2, A4 and A6
are their own inverses. The transformations A 3 and A5 are both
of order 3; and A2, A4, A6 all of order 2.
A set of g distinct linear transformations Si,..., so will
form a group of order g if the conditions of ~ 22 are all fulfilled.
Such a group we shall simply call a linear group.
84. Collineations and Collineation-groups. It is often convenient not to regard as essentially different two transformations whose matrices can be obtained one from the other by
multiplying all the elements of one by a constant factor, as, for
instance, in the case of
S 2 3 r4 6
s=, T=
0 1 0 2j
The two transformations are then said to represent the same
collineation. In other words, a collineation is specified by the
mutual ratios of the elements of the corresponding matrix,
not by the actual values of these elements. In practice it is
customary to affix a factor of proportionality to either the old
or the new variables to distinguish a collineation from a linear
transformation; the collineation represented by S or T above
would thus be written
pxi = 2x' +3x'2, pX2 = X2.
If A and B are two distinct linear transformations which
represent the same collineation, then BA-~=A-1B is a similarity-transformation. For, let the common ratio of the elements of the matrix of B to the corresponding elements of the
matrix of A be 0, then we immediately verify that B =AS =SA,
where S=(0, 0,..., 0).
If now a linear group G of order g be given, two cases may
arise. Either no two distinct transformations of G will represent the same collineation, or there will be some one set of,
say f, transformations which all represent the same collineation. In the former case G contains g distinct collineations;