CHAPTER IV. 
GROUP STRUCTURE AND SOME SPECIAL GROUPS. 
. Structure of the Collineation Groups of the Plane, 
. Singular Transformations. 
. Mixed Groups. 
. Generation of Finite from Infinitesimal Collineations. 
. The General Linear Group Gs(l~). 
. The Group G;(K). 
Exercises. 
UN Sn UR tn eR 
Doe WH eS 
In the last chapter we determined all varieties of collinea- 
tion groups in the plane and classified them with respect to 
the five types. In the present chapter we shall examine into 
the structure of each variety of continuous groups of plane 
collineations and discuss the generation of such groups from 
infinitesimal collineations. We shall also discuss in detail 
two specially important groups, viz.: G,(/~) and G,(K) and 
their subgroups. 
The structure of each variety of collineation group will be 
discussed in § 1, and the existence of the so-called singular 
transformations and their properties will be brought out in 
$2. Mixed groups of plane collineations, 7. e., groups not 
continuous but containing continuous subgroups, are treated 
in$38. $4 is devoted to the important topic of the genera- 
tion of continuous groups of collieations from infinitesimal 
collineations. The group G,(/), which leaves the line at 
infinity invariant, will be discussed in detail in $ 5, and the 
group G,(K) in § 6. 
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