432 Dr. Gr. A. Miller on the Substitution 



Later he gave a list of all the regular groups whose order 

 does not exceed 12 together with a geometrical representation 

 of them*. Kempe had previously given such a listf, but his 

 results were not quite correct. 



Since all groups are isomorphic to regular groups J and 

 two distinct regular groups cannot be simply isomorphic §, 

 it is clear that the enumeration of all such groups within 

 certain regions is very important. Complete enumerations 

 for the first part of the two following series of orders have 

 been published : (1) when the order of the groups is the 

 product of a given number of prime factors || , and (2) when 

 it does not exceed a given number If. 



Two more comprehensive enumerations with respect to 

 order may be mentioned, (1) the enumeration of all the tran- 

 sitive groups of given orders, and (2) the enumeration of all 

 the groups of given orders. The latter of these includes the 

 former, and each of them includes the regular groups. It 

 may happen that the transitive groups of a given order are 

 also regular. This is, for instance, the case when the order 

 is a prime number, or the square of a prime number. When 

 the order is a prime number (p) there is one group for every 

 degree which is a multiple of p : L e. there are n groups of 

 order p whose degree does not exceed np, n-1 of these are 

 transitive, n being any positive integer. It should be observed 

 that the number of the transitive groups of a finite order is 

 always finite, while that of the intransitive groups of any order 

 is infinite. 



When the order of the groups is a composite number, the 

 problem of determining all the possible groups becomes more 

 complex. We shall confine our attention to the groups 

 whose order is four. Since none of the transitive constituents 

 of these groups can be of an odd degree, we see that the 

 degree of such a group must be even and not less than four. 

 We may therefore represent the degree by 2n. 



To find all the cyclical groups of degree 2n we have 

 only to construct a 1,1 correspondence between a cyclical 



tran sitive groups/ a. ^^ J and a 2,1 correspondence between 



* American Journal of Mathematics, xi. (1889) pp. 139-157. 



t Phil. Trans, clxxvii. (1886) pp. 37-43. 



% Jordon, Traite des Substitutions, p. 60. 

 i § Netto, ' Theory of Substitution Groups ' (Cole's edition), p. 110. 



i| Holder, Mathematische Annalen, xliii. (1893) pp. 301-413; Cole 

 and Glover, American Journal of Mathematics, xv. (1893) pp. 191-221 ; 

 Young, ibid. pp. 124-179. 



If Miller, Comptes Rendus, cxxii. (1896) pp. 370-372, 



