GROUP STRUCTURES. 267 
GJ(AAV) = GI(AAV)+ HV) +H (AD, 
Gy (AA!) =01G/(AAV)+ H,(lA) +H, (1), 
Gi(W) = @1GJ(AAV)+ Hi(W) +H (A), 
GJ (AV) =@2G(AAV)+ Hs(V) +H (A) +H (UV) +G;/(AV), 
GJ (AN) = @?G/ (AAU) + F(A) +A (l) + H(A) + G3" (AT), 
Gi (Al) = 9G. (AAV) + ©? Hi (AU) + He! (A) + He! (1) + Gy! (Al). 
Synthetic Method. As another example, let us examine 
the structure of the group G,/(Al’). The point A’ may take 
co” different positions in the plane, and for each position of 
A’ there is a group G,'/(AA’l’) which belongs to the group 
G,/(Al’). The o* one-parameter perspective groups 
H,(A’,l’), contained in these groups G,/(AA’l’), belong to 
the group H,(/’). The group H,(l’) contains the subgroup 
H,/(l'). The ~? groups of elations H,'(Al), contained in 
the groups G,/(AA’l’), form the group H,'(A) which is, 
therefore, contained in G,/(Al’). Since G,/(Al’) contains 
H,/(A) and G,'(1), it must also contain G,’(Al’). Thus 
the structure of G,/(Al’) is found. A similar course of rea- 
soning leads to the structural formule of the other groups of 
this class. 
Analytic Method. The normal form of a collineation T of 
type II in G,/(Al’) is, Art. 137, 
Bi 
i y+ Tie 1) x 
he iar i / 
, At == = — Sass _ 
- f foal Jaf » ¥, ‘jp=a [eh (4) 
i+ty+(—— _—4 )e SEU) se -—t)a 
AY Al A! Al 
If t = 0 in equations (4) these become 
B’ 
Y= (la) 
: kx A’ 
6 Ras eee Bot 2 (4a) 
1 foe fr 
which are the equations of the group H,(l’). 
If k=1 and A’ =0 in (4a) while tim. —- = t’, then these 
equations reduce to 
x y+ Btls 
%,=——, Y,= —- 
oe hit afi Yi 1+t/x 
These are the equations of the group H,'(1’). 
(4b) 
