NOTE ON THE PROJECTIVE GROUP, 

 By E. W. Rettger. 



Presented by Henry Taber, April 13, 1898. 



TiiK general projective group occupies a position of special importance 

 in Lie's theory of finite continuous groups. For associated with any 

 finite continuous group G^ with r parameters is a sub-group with p^r—\ 

 parameters of the general projective group in (r — l)-fold space, the 

 knowledge of whose invariants (general and special) enables us to enu- 

 merate the different types of sub-groups of Gr- This projective group is 

 obtained from the adjoined group of Gr by regarding the variables in the 

 ecpiatious of transformation of the adjoined as homogeneous co-ordinates. 



Lie showed that the general projective group is continuous, in the 

 sense that each transformation of this group can be generated by an 

 infinitesimal transformation of the group.* But Professor Study made 

 the important discovery that not every transformation of the special 

 linear homogeneous group can be generated by an infinitesimal transfor- 

 mation of the special linear homogeneous group ; f and thus showed that 

 the sub-groups of the general projective group are not all continuous in 

 the sense in which this term is here employed: namely, a group is here 

 termed continuous if each transformation of the group can be generated 

 by an infinitesimal transformation of the group, and therefore belongs to 

 a continuous one-term sub-group of the group in question. 



Subsequently Professor Taber showed that not every transformation 

 of the orthogonal group in n variables, for n > 4, can be generated by 

 an infinitesimal transformation of this group, and established equivalent 

 results for the group of autoniorphic linear transformations of an alternate 

 bilinear form, and for the group of autoniorphic linear transformations of 

 a general bilinear form.t I have found, moreover, that a number of tlie 



* Lie, Contiiiiiid-'iche Gruppen, p. 45. 

 t Leipzige Berichte, 1892. 



t Bii/I. N. Y. Math. Societi/, July, 1894; M,it/>. Ann., Vol. XLVI. p. 5G1 ; Math. 

 Review, Vol. I. p. 154. 



