166 THEORY OF COLLINEATIONS. 
angle of a two-parameter subgroup of G,(A). The groups 
G,(l) and G,(A) are dualistic groups. G, contains «* sub- 
groups G,(l) and also ~* subgroups G,(A). 
Each of these groups contains, besides collineations of type 
I, also collineations of other types ; but the collineations com- 
posing a group G,(l) or G,(A) are chiefly of type I. The 
structure of all these groups will be examined in detail 
in Chapter IV. 
200. The Group G,(Al). The general projective group 
G, contains ~* subgroups G,(Al), one for each lineal element 
in the plane. Each group G,(Al) is made up of «* two-par- 
ameter groups G,(AA’A”) whose invariant triangles have 
one vertex and one side in common. Each group G,(l) con- 
tains ©’ subgroups G,(A/), one for each point on /; likewise 
each group G,(A) contains ~’ subgroups G, (Al), one for each 
line through A. The group G,(AL) is self-dualistie. 
201. The Groups G,(AA’') and G,(Ill’). Two points AA’ 
and their join / form the invariant figure of a four-parameter 
group G,(AA’). The group G, contains ~/ equivalent sub- 
groups G,(AA’), one for each pair of points in the plane. A 
group G,(AA’) is composed of «’ two-parameter groups 
G,(AA’A"’) whose invariant triangles have the vertices A 
and A’ in common. 
In like manner, G, contains ©* equivalent subgroups G, (Il’), 
one for each pair of lines in the plane. A group G,(ll’) is 
composed of ©” two-parameter groups G,(AA’A’’) whose in- 
variant triangles have one vertex and two sides in common. 
The groups G,(AA’) and G, (/l’) are dualistic. G,(1) contains 
o’* subgroups G,(AA’), also ~* subgroups G,(ll’); G,(A) 
breaks up in a similar manner. 
202. The Group G,(A,l’’). There are ©* combinations of 
point and line in a plane where the point is not on the line. 
Each combination of point and line is the invariant figure of 
a four-parameter group G,(A,/’). The group G,(A,l’) is 
composed of ” two-parameter groups G,(AA’A’’), whose in- 
