242 MILLER— GROUPS GENERATED BY TWO OPERATORS. 



direct product of H and this dihedral group. As these two groups 

 clearly generate G it results that G is this direct product. That is, if 

 tzvo operators of order i8 satisfy the conditions Sj''^s/Sj^=^So'*, 

 Sz'^Sj^s = s-^''^ they generate the direct product of a dihedral group 

 and a certain non-ahelian group of order 27. Moreover, every such 

 direct product can he generated by two operators of order 18 satis- 

 fying the given conditions. 



If only one of the two operators s^, S2 is of order 18 the other 

 must be of order 9 since s.^-, s^' cannot be commutative. That is, 

 if an operator of order 18 and an operator whose order is not 18 

 satisfy the conditions s-^'^S2^s^ = Sn'^, Sn~'^S{^So = sf^ these operators 

 must generate the direct product of the group of order 2 and 

 a certain group of order 27. When .jj, Sn are both of order 9 

 they generate a group of order 27, and when both are of order 2 

 they generate a dihedral group. Combining these results we have 

 that if two non-commutative operators satisfy the two conditions 

 sf^s^^s^ = sf*, s.f^s^^s^ = s^-* their orders have one of the follow- 

 ing pairs of values 18, 18; 18, 9; 9, 9; 2, 2. In the first and last 

 of these cases G may be any one of an infinite system of groups ; 

 viz., any dihedral group in the last case, and the direct product of 

 such a group and a certain group of order 27 in the first case. In 

 each of the other two cases there can be only one group, viz., the 

 non-abelian group of order 27 which involves operators of order 

 9 in the third case, and the direct product of this group and the 

 group of order 2 in the second case. 



University of Illinois, 

 December, 1909. 



