124 The Transitive Substitution Groups of Order 8p. 



For the sake of completeness we shall consider these groups 

 very briefly, although all of thern are known. 



Since 4 ! is not divisible by 16 it follows from theorem I. 

 that every subgroup of order 4 contained in a group of order 

 16 must include an invariant subgroup which differs from 

 identity. All the subgroups of order 8 contained in such a 

 group are known to be invariant. It remains therefore only 

 to consider the subgroups of order 2. 



We may now show directly by means of the groups of 

 order 8 that an operation group of order 16 cannot be simply 

 isomorphic to more than one non-regular transitive group. 

 We may regard a commutative subgroup of order 8 as the 

 head of such a group. When this is cyclical the truth of 

 the statement is evident, since any operation of order 2 in the 

 tail may be made to correspond to any other of its operations 

 of this order. When the head contains four operations of 

 order 4 and an operation of the tail is commutative to its three 

 operations of order 2, the statement is true for the same 

 reason. Finally, when the tail contains no operation that is 

 commutative to the three operations of order 2 in the head, 

 and also no operation of order 8, it must contain 4 operations 

 of order 4 and 4 of order 2, since any operation of the tail 

 and the subgroup of the head which is generated by its 

 operations of order 2 must generate the non-commutative 

 group of order 8 which contains five operations of order 2. 



In this last case we have that an operation of order 2 in 

 the tail multiplied into two operations of order 4 in the head 

 gives two operations of order 2. Hence the operations of the 

 tail transform one of the cycles of order 4 in the head into 

 its third power, and are commutative to the other two opera- 

 tions of order 4 in the head. Hence any operation of order 

 2 in the tail may be made to correspond to either of the two 

 non-commutative operations of this order in the head as well 

 as to any other operations of the same order in the tail. As 

 the only group which does not contain at least one of the two 

 given commutative groups of order 8 is commutative, the 

 statement is proved. 



We have now proved that every operation group of order 

 16 that contains non-commutative operations of order 2 is 

 simply isomorphic to one and only one non-regular transitive 

 group. Hence there are 20 transitive substitution groups of 

 order 16 ; 14 of these are regular. 



Summary. 

 When j9 = 2 there are 20 translative substitution groups of 

 order Sp. Six are of degree 8 and fourteen are of degree 16. 



