GROUPS AND MANY-VALUED FUNCTIONS. 319 



468152739 312569478 759618342 

 231698547 975182634 846529173 

 846371925 231745896 975436281, 



where the two derived groups are not derived derange- 

 ments. 



It is not easy to give an exact enumeration of the 

 grouped groups formed by auxiliary groups g which have 

 any circular factors whatever made with the a elements, 

 the h elements, &c. 



39. Perhaps it may simplify the conception of these 

 grouped groups if we remark that in any group g which 

 we select as the auxiliary group, as, for example, 

 123 213 

 231 132 



312 321, 

 we may substitute for the elements 123 any square groups 

 of m powers of the same number m of elements ; that is, 

 we may write above in g throughout 



for 1, 123 ; for 2, 456 ; and for 3, 789 

 231 564 _ 897 



312 645 978, 



and the result will be a grouped group. 



It is evident that if we write the exponent 2 over the 

 element 3 in g, the result 



123'- 



23^1 



3^12 



1 3^2 



3^2 1 



2 1 f 



iff') 



is still a group made with the three elements 123^; and 

 the substitutions Q1Q.2Q3Q4Q5 above formed have all 123^, 

 neglecting the p and reading only subindices and expo- 

 nents, in the denominator, while their numerators show 

 the remaining substitutions of {g'). It is evident also since 



