30 Two- AND THREE-PARAMETER GROUPS. 
If T belongs to G, it is evident that G’ is the same group 
as G; for the resultant of T and S is always a transforma- 
tion in G and the resultant of 7‘ and (ST) is also in G. If 
T does not belong to G, then G and G’ are usually not the 
same group, but may be the same in some cases. 
If G is the general projective group in one dimension and 
T is also a projective transformation, then G’ is the general 
projective group. If Gis some subgroup of the general pro- 
jective group and T does not belong to G, then the trans- 
formed groups G’ and G are said to be equivalent subgroups. 
The invariant figure of G is tranformed by T into the invari- 
ant figure of G’ and corresponding transformations in G and 
G’ have the same cross-ratio; the two subgroups G and G’ 
have, therefore, the same structure. 
36. Invariant Subgroup. When a group G is trans- 
formed by T into G’, the subgroups of G go over into the 
subgroups of G’. When T belongs to G, the subgroups of G 
are only interchanged, since G’ is the same as G. If a sub- 
group of G not containing 7 is transformed by T into itself, 
such a subgroup is called an invariant subgroup of G. 
As an example let us operate on the group G.( A) by any 
transformation T belonging to G,(A). Take S in the form 
ax + dy cx + d~ 
found by making b = b, = 0, in equation (28). Thus 
o— and T in the same form «, = S’is readily 
Ss’: aloe CE dea . (39 ) 
Since S’ is also in G,(A), it follows that 7 has transformed 
G.( A ) into itself. 
G,(A) contains one subgroup of type II, viz., G,/(A). If 
axe + ay 
S be chosen from this group, its equation becomes x, = 
adi & 
S’ then reduces to #, = whence S’ also belongs to 
dea + aa’ 
G,/(A). Hence G,’( A) is an invariant subgroup of G,’'(A ). 
