GROUPS AND MANY-VALUED FUNCTIONS. 283 



derangements of the same equivalents, including the equi- 

 valents ; for every derivate of Gi is a derangement of G2 

 equivalent to Gj . 



If the number of groups equivalent to G is less than 

 that of its derived groups, we know that G has derived 

 derangements. If M be the number of groups equivalent 



JTN 



to G, including G, -nrjr is the number of derived derange- 



° km ° 



ments of G or of any equivalent to G (including G and 

 the equivalent) ; and each of the equivalent groups forms 



TIN 



with its derived derangements a modular group of -^^ 



substitutions. 



Def. A modular group consists of a model G and certain 

 derived derangements of G. A non-modular group is a 

 model group which cannot be written as a model with its 

 derived derangements. 



§ 3. 

 Model groups of the order k, made with N elements, which 

 are the powers of a substitution of the k*^ order. 

 10. Let 



N=Aa + BZ> + Cc+..+J/ 

 be any partition of N, such that 



A>B, B>C, &c., J>o, 

 abc • 'j being any numbers ; and let k be the least common 

 multiple of ABC • • J. 



We are about to construct a group of the powers of a 

 substitution having a circular factors of the order A (of 

 A elements), b of the order B, &c. 



The elements of the first factor of the order A may be 

 selected among the N elements 123- -N in 



TIN 



HA n(N-A) 

 different ways. 



