GROUPS AND MANY-VALUED FUNCTIONS. 379 



is (L) = 1X1X2^3 • • where W • • are all of the p*^ order. 

 The substitutions 



^1/^2/^3 • 'H'N-l 



of any of the derived derangements of (L) are such that 

 we have always 



wherefore 



is a substitution of the derivate, which is similar to /*„; 

 and the derivate is composed of N - 1 similar substitu- 

 tions. 



The group of M. Mathieu, 



K=LM„, 

 of (N-i)(N-2) substitutions made with the elements 

 123 • • (N - 1)^ which we suppose to be given, is 



which can be written also as the product 



K=L.Mi.M2.M3..M;v-i, 



where M^^ is the group of (N - 2) powers of f^^ . 



JV-2 N — 2 . . 



The derivate /i„~2~ L, when is integer, that is, 



when p>2, is always a system of didymous radicals of 



Let the N-2 didymous radicals of M^ be written below 

 M^. We have now N- 1 model groups 



M'lM'aM's.-MVi 

 each of the order 2(N-2)j and each one having"a different 

 element undisturbed in a vertical row. 



Let N final be added to every substitution of K, and in 

 MV so augmented let N be exchanged in all the N-2 

 didymous radicals with the undisturbed letter r. The 

 result is M"^ which is still a group of N - 2 powers with 

 N-2 didymous radicals. 



