Two- AND THREE-PARAMETER GROUPS. 29 
that divide A’ and A harmonically, it follows that there are 
co’ involutoric transformations that interchange A’ and A. 
The system of transformations in mG,(AA’) possesses both 
group properties. This is known to be true for these trans- 
formations in mG,(AA’) which belong to the continuous 
group G,(AA’); but it must be proved for those transforma- 
tions that interchange A’ and A. Let T and T, be two in- 
volutoric transformations each interchanging A’ and A; their 
resultant, therefore, leaves both A and A’ separately invari- 
ant, and hence belongs to the continuous group G,(AA’) and 
is also in mG,(AA’). Let T and T’ be two transformations, 
the first leaving A’ and A separately invariant, and the 
second interchanging A’ and A. Their resultant interchanges 
A’ and A, and is therefore an involutoric transformation be- 
longing tomG,(AA’). Hence all transformations in mG,(AA’) 
have the first group property. Since every involutoric trans- 
formation is its own inverse (art. 27), it follows that mG,(AA’) 
has the second group property. Hence it is appropriate to 
call the set of transformations in mG,(AA’) a mixed group. 
THEOREM 18. The aggregate of those transformations inter- 
changing a pair of points and those leaving them separately invari- 
ant forms a mixed group mG:( AA’). 
35. Operation by Ton G. If we operate with T as in art. 
24 on all the transformations of agroup G, we produce thereby 
a new group G’. This is proved as follows: Let S and S, be 
any two transformations of G and let SS, = S,, whence S, is 
also a transformation inG. Operating with 7’ on S and S, we 
et 
Se iS ieands. — Las ws 
hence SS le Ne Se — er Sls — ee Se, 
or SS 
i. e., the result of operating with T on the resultant of S and 
S,, is the resultant of S’and S,’.. Therefore if the transforma- 
tions S form a group G, the transformations S’ form a new 
group G’. We express this by saying that 7 has transformed 
the group G into G’, = 
