I507.1 MILLER— GROUPS OF ORDERS TWO AND FOUR. 147 



and consider the four operators 



The common product of the two commutators and of the other two 

 factors is s^^s^sj^s-^s^^. Hence each of these four factors transforms 

 this common product into its inverse. Moreover, the common 

 product of the first and third and of the second and fourth of the 

 four given operators is s-^^s^-s-^^s^^, and each of these operators trans- 

 forms this common product into its inverse. From this it follows 

 that the two operators 



So^s^So'S-^So^ and s^s^'S^s^^ 



are commutative and that the group generated by them is either 

 cycle or the direct product of two cyclic groups. This abelian 

 group, together with s^s^^s^s^, generates a group (H), which in- 

 volves Sj^s^s-^s^^, S2^Sj^s./SiS2-, s^^s^s^^Sj^s^^. The order of H cannot be 

 more than twice that of the given abelian subgroup, and H is invar- 

 iant under the group (G) generated by s^ and So, since Jo transforms 

 the four generators of H among themselves and s^ transforms the 

 first two of them into themselves, while 



From theorem II we conclude that H is the commutator sub- 

 group of G and we have the preliminary theorem : If a group is 

 generated by two operators of orders 2 and 4, respectively, whose 

 commutator is of order 2, the commutator subgroup must be of 

 one of the following two types : the cyclic group of order 2, the 

 group obtained by extending a cyclic group or the direct product 

 of two cyclic groups by means of an operator of order 2 which 

 transforms each operator of this direct product into its inverse. 



The group generated by H and ^i is invariant under s., since it 

 contains s^^s^So. Hence the order of G cannot exceed eight times 

 that of H. As the operators of odd order in G (if such operators 

 occur) generate a characteristic subgroup of its commutator sub- 

 group the order of the commutator quotient group is either 4 or 8; 

 that is, the order of G is either four or eight times that of H. 



