150 MILLER— GROUPS OF ORDERS TWO AND FOUR. [April 19, 



follows that ^o"and this direct product generate H, and hence the 

 order of G is S/?-. We have thus arrived at an interesting infinite 

 system of transitive groups of degree 2p and of order 8/>-, such that 

 each group is generated by two substitutions of orders 4 and 2, re- 

 spectively, whose commutator is of order 2. 



As an indirect result of the preceding paragraph we have that 

 the group of isomorphisms of the abelian group of order p- and 

 type (i, i) must always include the octic group. As the abelian 

 group of order />"^ m>2, and of type (i, i, i, . . .) can be made 

 simply isomorphic with itself after any correspondence has been 

 established between two of its subgroups, it follows from the above 

 that the octic group is a subgroup of the group of isomorphisms 

 of the abelian group of order p^, w > i, and of type (i, i, i, • . •), 

 p being any odd prime number. In particular, this group of iso- 

 morphisms contains two operators of orders 2 and 4, respectively, 

 whose commutator is of order 2. 



The main results of the preceding paragraphs may be stated as 

 follows : If G is generated by two operators of orders 2 and 4, re- 

 spectively, whose commutator is of order 2, then G is solvable and 

 has an (a, i) isomorphism with the octic group. The commutator 

 ciuotient group of G is of order 4 or 8. When this order is 4 the 

 order of G is Sm, m being an odd number, and vice versa. The 

 highest power of an odd prime factor of the order of G has an even 

 index and the Sylow subgroup of this order is the direct product 

 of two cyclic subgroups of the same order. When the commutator 

 subgroup of G is cyclic G is either of order 8 or of order 16 and 

 vice versa. The only other abelian commutator subgroups which 

 may occur in G are the four-group and the abelian group of order 

 8 and of type (i, i, i). The other possible commutator subgroups 

 of G may be obtained by extending 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. A cyclic subgroup 

 whose order exceeds 2 which is contained in this direct product is 

 not invariant under G. 



University of Illinois, 

 January, 1907. 



