1907J MILLER— GROUPS OF ORDERS TWO AND FOUR. 149 



which have been obtained may be expressed as follows: // a group 

 is generated by tzi'o operators of orders 2 and 4, respectively, whose 

 commutator is of order 2, the commutator subgroup is either the 

 cyclic group of order 2, or it may be constructed by extending the 

 direct product of tzvo cyclic groups, involving the same odd factors, 

 by means of an operator vf order 2, zvhich transforms each operator 

 of this direct product into its inverse. 



It should be observed that the Sylow subgroups whose orders 

 are powers of 2 in the factors of the direct product of the preced- 

 ing theorem are either of the same order or one of these orders is 

 twice that of the other. In particular, this direct product may be 

 of order 2. In this case H is the four-group. When H is abelian 

 its order can clearly not exceed 8, and when it involves operators 

 of odd order its order cannot be less than 18. The octic group and 

 the given group of order 16 are evidently the only possible groups 

 when H is cyclic and the substitution group of order 64 and degree 

 8 which may be generated by s-^ = ae, ^2 = abcd-efgh is a group 

 in which G has its largest possible value when H is abelian. As an 

 instance of a G which involves operators of odd order we may men- 

 tion the transitive group of degree 6 and of order y2. This may be 

 generated by ^^ = ae and s., = abed • ef, and is the smallest possible 

 group involving operators of odd order, which may be generated 

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

 is of order 2. 



Let ^o be a substitution of degree 2p, p being any odd prime 

 number, involving only one transposition. Representing s^ by 

 a^aoOM,^ • b^bob^b,^ .... l-JJl.K • ^''^^ ^^^^ s^ by a^b^ ■ b^c^ .... l^m, it is 

 clear that s^, s. generate a transitive group of degree 2p. As the 

 order of H is divisible by p^ and as it contains a direct product of 

 two cyclic groups whose orders are divisible by p, this direct product 

 must be of order p~. The order of H is therefore 2p- and H is 

 obtained by establishing a {p, p) correspondence between two dihe- 

 dral groups of order 2p. Since ^o- is not commutative with any 

 substitution in the direct product of order p- contained in H besides 

 the identity, it has />- conjugates under this direct product and hence 

 transforms each one of its substitutions into its inverse. From this it 



