22 ONE-PARAMETER GROUPS. 
complex number system, its reciprocal, 1 /k, is also a number 
in the same system. Hence the inverse of every transforma- 
tion in the group G,(AA’) is also in the group. This is called 
the second group property. 
The resultant of a pair of inverse transformations is the 
identical transformation, whose cross-ratio is given by 
kX1/k=1. Hence the group G,(A A’) contains the identical 
transformation. 
The group G,(AA’) contains one transformation which is 
identical with its own inverse. In this case we have the con- 
dition k = = or k#?=1; whence k==+1. The value k=1 
gives the identical transformation of the group. That this is 
its own inverse is self-evident. The value k= —1 gives the 
involutoriec transformation of the group. This transforma- 
tion has the effect of interchanging every pair of correspond- 
ing points on the line, since its second power is the identical 
transformation; thus this transformation gives rise to an in- 
volution, whence its name. 
The group G,(AA’) contains two very noteworthy trans- 
formations whose cross-ratios are 0 and, respectively. The 
first transforms all points of the line except A’ into A; the 
second transforms all points of the line except A into A’. 
These are pseudo-transformations and may be regarded as 
forming an inverse pair. 
The cross-ratio of the identical transformation is unity, and 
this transformation leaves every point of the line invariant. 
The transformation of the group whose cross-ratio is 1+ 4, 
where 4 is an infinitesimal number, moves every point on the 
line an infinitesimal distance, and hence is called an infini- 
tesimal transformation. 4 has an infinite number of dif- 
ferent values, viz., |p| e“, where is an infinitesmal and 6 
varies from 0 to 2x. If an infinitesimal transformation be 
repeated ” times, the cross-ratio of the resultant is (1+ 4)”. 
By a proper choice of 4, 7. e., of 6, and of x (sufficiently large), 
this cross-ratio may be made any number we please; hence 
