Groups whose Order is Four, 435 



spond to a particular value of w, we may therefore first find 

 the sum of the number of these groups which correspond to 

 the different values of S that satisfy the relation 



= 2n 



3 ' 



S<- 



and deduct from this sum the sum of \ -pr — hr terms of 



ffl-G-r 



the corresponding series (A) . These operations are indicated 

 in the following formula : — 



JEM B]([! ] +2 ) H «— >(BG-i )([t]-2- +1 ) 

 few 01 *)(G ] +2 ) - 1 11 ^(KHI]) 2 - 



By means of this formula we can readily determine the 

 number of groups which contain 2n elements and n systems 

 of intransitivity for any particular small value of n. The 

 individual groups may be found by assigning to S the 



successive 



integers from 1 to -o- and rejecting the identical 



groups according to series (A) when the value of S satisfies 

 the relation 



The groups of the second type of non-cyclical groups, 

 which correspond to the other values of a, are found in 

 exactly the same way. Their number may therefore be 

 found by means of the given formula provided we use instead 

 of n the following series in order, 



n— 2, Ti—4, Ti—6, . . ., 2 or 3. , 



By adding the double of the largest value of a to the sum of 

 the numbers of these groups corresponding to the different 

 possible values of a, we obtain the number of groups whose 

 order is four and whose degree is 2ti. Two of the groups 

 are transitive when n is 2. For the other values of n all the 

 groups are intransitive. 



Example, 



It is required to find all the groups whose degree and 

 order are 14 and 4 respectively. 



* The brackets indicate that the largest integer which does not exceed 

 the inclosed fraction is to be used. 



2H2 



