148 MILLER— GROUPS OF ORDERS TWO AND FOUR. [April ,9, 



The octic group is clearly the group of smallest order which 

 contains two operators of orders 2 and 4, respectively, whose com- 

 mutator is of order 2. In this case the commutator subgroup is 

 cyclic and the commutator quotient group is the four-group. There 

 is one group of order 16 which is generated by two operators of 

 orders 2 and 4, respectively, whose commutator is of order 2. This 

 may be defined as the group of order 16 which involves the abelian 

 subgroup of type (i, i, i) and 4 cyclic non-invariant subgroups of 

 order 4. In this case the commutator subgroup is again cyclic and 

 the commutator quotient group is of type (2, i). Since this group 

 of order 16 has a (2, i) isomorphism with the octic group, and 

 since the operators of odd order in G generate a characteristic sub- 

 group, it follows that every group zvhich is generated by two opera- 

 tors of orders 2 and 4, respectively, zvhose commutator is of order 2, 

 has the octic group for a quotient group, and its commutator quo- 

 tient group is either of type (i, i) or of type (2, i). 



Suppose that the commutator quotient group of G is of order 4. 

 If the order of G were divisible by 16 it would have the group of 

 order 16 considered in the preceding paragraph as a quotient group. 

 This is impossible since the commutator quotient group of the latter 

 is of type (2, i). That is, if the commutator quotient group of G 

 is of order 4 the order of G is 8 m, m being an odd number. 



If H contains an invariant cyclic subgroup whose order exceeds 

 2, the operators of H, which are commutative with a generator of 

 this subgroup, constitute an invariant subgroup of G. This invar- 

 iant subgroup does not include s^^So^s^So, and hence it does not in- 

 clude H. Since the group of isomorphisms of a cyclic group is 

 abelian and every invariant subgroup which gives rise to an abelian 

 quotient group includes H, the assumption tliat H contains an invar- 

 iant cyclic subgroup whose order exceeds 2 has led to a contradic- 

 tion. From this it follows that the groups generated by So^s^^s^-s^^So^ 

 and Sj^s/s^s./, respectively, have at most 2 common operators and 

 that these cyclic groups are either of the same order or the order 

 of one is twice that of the other. In particular, if the order of G 

 is divisible by an odd prime number p, the highest power of p 

 which divides this order has an even index. The main results 



