164 THEORY OF COLLINEATIONS. 
type I in common; for, if two collineations T and T, are 
the same, the eight parameters of the one must be equal to 
the eight parameters of the other; but if the collineations 
leave different triangles invariant, all of the six coordinate 
parameters of the one cannot be equal to those of the other. 
Hence 7 and JT, cannot be the same except when both are 
identical collineations. 
198. Subgroups of G,. By the aid of the principle that all 
collineations which leave a certain figure invariant form a 
group, it is not difficult to enumerate all varieties of sub- 
groups of G, that can be compounded out of the * two-para- 
meter groups G,(AA’A’’) in the plane. It is only necessary 
to recall all configurations of lines and points that can be 
made up of triangles. They are as follows: a line l, a point 
A, a pair of lines //’ (and their intersection ), a pair of points 
AA’ (and their join), a lineal element Al, a point A anda 
line / not through A, two points AA’ their join and a line / 
through one of the points, three points or three lines forming 
a triangle. These eight configurations (shown in Fig. 24) are 
the invariant figures of subgroups of G,, as follows: G,(l), 
G,(A), G,(Al), G(AA’), GU’), G,(4,l"), G (AAT), 
G,(AA’A”). We shall briefly discuss each variety of group 
in detail. 
199. The Groups G,(l) and G,(A). There are ~* col- 
lineations of type I in the plane and only ~®* lines in the 
plane; hence, any line of the plane can be transformed into 
any other line or into itself in ~° different ways. The o° 
collineations, which transform a line / into itself, form a six- 
parameter group G, (1). There are ~‘ triangles having the 
side 1 in common; each of these triangles is the invariant 
triangle of a two-parameter group G,(AA’A”). Thus we see 
that G, (1) is made up of ~* two-parameter groups G,(AA’A”). 
In like manner, we see that any point A of the plane is the 
invariant figure of a six-parameter group G,(A). Each of 
the / triangles having A for one vertex is the invariant tri- 
