Groups whose Order is Four. 433 



each one of these groups and a group of the second order 

 whose degree is 2ra — 4a. The number of such groups for a 

 given value of n is therefore equal to the largest value of a, 

 and the individual groups may be given by assigning to a the 

 successive integers beginning with unity. 



To find all the non-cyclical groups of degree 2n we may 

 construct a 1,1 correspondence between a four-groups*, and 

 (1) a 2,1 correspondence between each one of these groups and 

 a group of the second order whose degree is 2n — 4a, (2) a 

 1,1 correspondence between each one of these groups and a 

 group of the fourth order which is of degree 2/i — 4a and 

 contains n — 2a systems of intra nsitivity. The number of the 

 groups of the first one of these two types is the same as that 

 of the cyclical groups, and the individual groups may be 

 given in the same way. The number of groups of this and 

 the cyclical type is therefore twice the largest value of a. 



The groups of the second type of non-cyclical groups 

 present somewhat greater difficulties. Here a may assume 

 the value zero in addition to its values in the two preceding 

 cases. We shall first determine the groups when a is zero ; • 

 i. e., we shall first seek all the 



Groups which contain 2n Elements and n Systems of 

 Intransitivity . 



The average number of elements in the substitutions of 

 such a group is nf, and the number of elements in all of its 

 substitutions is 4w. The number of systems of two elements 

 is therefore 2n. These 2n systems must occur in three sub- 

 stitutions. If the smallest number of systems in any one of 

 these three substitutions is represented by S, we have 



For each value of S which satisfies this relation there must 

 be at least one group, since we have only to use the remaining 

 systems for the second generating substitution in order to 

 construct such a group. 



In general we have the following : — 



* Bolza, American Journal of Mathematics, xi. (1889), p. 297. 

 t Frobenius, Crelle's Journal, ci. (1887) p. 287. 



Phil Mag. S. 5. Vol. 41. No. 252. May 1896. 2 H 



