SLOCUM. — FINITE CONTINUOUS GROUPC. 101 



where A^ v is the first minor of A relative to G , and thus the A's are 

 integral functions of a x , a» . . . a,.. Consequently, if A £ 0, the com- 

 position of the transformation e a and an arbitrary infinitesimal transforma- 

 tion e lfi gives a transformation e a+ ty , infinitely near the transformation 

 e a , and generated by an infinitesimal transformation. Let e a ~^~ siy be 

 denoted by e" 1 , that is, let e" 1 = e a ' Y , where 



ax = a + 8 t y = a : X 1 + a., X 2 + . . . + a,. X r . 



Applying the infinitesimal transformation e repeatedly, we thus obtain 

 the equations 



e °» = e a "-V^ = e°V 5//3 . 



For n infinite, «8< is finite, and may be taken equal to unity ; thus 



a„ a (3 



e " = e e . 



Consequently, if A does not vanish for any system of values of a x . . . a r , 

 in which case A is a constant,* then the composition of an arbitrary 

 transformation e a with finite parameters with an arbitrary transforma- 

 tion e" = e with finite parameters, gives a transformation of the group 

 with finite parameters which is generated by an infinitesimal trans- 

 formation. 



The form of A depends only on the structural constants, and thus A is 

 the same for all groups of the same structure. Therefore, if the A cor- 

 responding to a given structure is a constant, the composition of two 

 arbitrary transformations of any group of this structure gives a transfor- 

 mation of the group with finite parameters, that is, a non-singular 

 transformation of the group, and consequently every group of this 

 structure is continuous. | 



If the A corresponding to a given structure vanishes for certain systems 

 of values of a t ... . a r , some groups of this structure may be continuous 

 and others discontinuous. For example, the two groups G 2 and G 2 , con- 

 sidered above, page 92, both have the structure (X 1} X 2 ) = X l . The 



* For complex groups A is either unify, or else vanishes for certain systems of 

 values of «! . . . a r . See the expression for A as a product on page 104. 

 t This criterion of continuity is due to Professor Taber. 



