SO^IE RELATIONS BETWEEN SUBSTITUTION GROUP 

 PROPERTIES AND ABSTRACT GROUPS. 



By G. a. miller. 



{Received June 4, 1910.) 



Dyck observed that the necessary and sufficient condition that a 

 transitive substitution group G of degree n is primitive is that its 

 subgroup Gi which is composed of all the substitutions of G which 

 omit one letter is a maximal subgroup of G^. This establishes an 

 important relation between primitive substitution groups and the 

 properties of G considered as an abstract group. It has also been 

 observed that abstract group properties correspond to the different 

 degrees of transitivity of substitution groups-, and that the number 

 of ways in which an abstract group can be represented as a transi- 

 tive substitution group can be directly deduced from the properties 

 of the group. These and other known relations have done much 

 towards uniting the theories of abstract groups and those of sub- 

 stitution groups, and towards making these theories mutually help- 

 ful. It is the object of the present paper to establish other im- 

 portant relations between these two theories, especially as regards 

 abstract groups and simply transitive substitution groups. 



§ I. Sonic properties of co-sets. 



li H represents any subgroup of index p under a group G all the 

 operators of G may be arranged so as to give p distinct sets both as 

 regards right-hand and as regards left-hand multiplication, in the 

 following forms :^ 



G = H-^HS,-\-HS,^. . .-^HS^, 

 = H-\-SM + S,H -f . . .-{-S,H. 



^ Mathcmatische Annalen, vol. 22 (1883), p. 94. 



'^Messenger of Mathematics, vol. 28 (1899), p. 107. 



^Bulletin of the American Mathematical Society, vol. 16 (1910), p. 454. 



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