GROUPS GENERATED BY TWO OPERATORS EACH OF 



WHICH TRANSFORMS THE SQUARE OF THE 



OTHER INTO A POWER OF ITSELF. 



By G. a. miller. 



{Read April 23, 1910.) 



Two special cases of the category of groups defined in the head- 

 ing of this paper have been considered, viz., when the square of 

 each of the two generating operators (s^, s^) is transformed either 

 into itself or into its inverse by the other. ^ In each of these cases 

 it was found that the orders of s-^, s^ do not have an upper limit. 

 It will be found that both of these orders must always have an 

 upper limit unless at least one of these operators transforms the 

 square of the other either into itself or into its inverse. 



The given conditions give rise to the following equations: 



If at least one of the two numbers a, b were odd the order of the 

 corresponding operator would be odd, and hence the group (G) 

 generated by s^, s^ could be generated by a cyclic group of odd 

 order and an operator transforming this group into itself. As 

 many of the properties of such a group are known we shall confine 

 our attention in what follows to the cases when both a and b are 

 even numbers, and hence we shall assume that the conditions under 

 consideration are written in the form 



Sf'^So'S-^ = .Jo^" , ^2~^«^l'-^2 = ^ 



2/5 



Some fundamental properties of .y^, So may be deduced from 

 these equations in the following manner: 



f -2^ 2 2 _ 2a2 C -2c 2c 2 = c 23= c -2c -2^ 2 2^a"—1^^ -2 



•^1 •^2-'l -^2 ' 2 -^1 "^2 ■*! ' •'2 •*! -^2 '^2 "^l ' 



c -232 _ f 2(a2-l)^. -2 ^ 2(32_i) 2(a2-l) _ t 



*Cf. Paris Comptes Rendiis, Vol. 149 (1909), p. 843. 



238 



