GROUPS AND MANY-VALUED FUNCTIONS. 341 



IIN 



derived KG. and the function ^ has fewer than -^^r— 



Jj 



values. Q.E.D. 



57. If we take a maximum group G consisting of a 

 nucleus group g and all its derived derangements, and if 

 we select a system of exponents of which some are equal 

 to others^ we have only to consider how many of the 

 groups equivalent to G give the same algebraic function 



The number of these equivalents is always given, if G 

 be any of the groups constructed by the theorems which 

 have preceded; and there is no difficulty in determining 

 the groups which will give the same function. 



Two equivalent groups which give the same ^, and 

 which have e nonrepeated exponents, can always be so 

 arranged that the e vertical rows under those exponents 

 shall be identical in the two groups, and that the tactical 

 difference between the groups shall be limited to vertical 

 rows under repeated exponents. 



Let the system of N exponents chosen be a a times, /5 

 h times, 7 c times repeated, &c,, giving 

 N=aa + &/3 + C7H 



We can readily satisfy ourselves, by inspection of the 

 equivalent groups 



G, AGA-\ BGB-i.-, 

 or by consideration of the law that governs them, how 

 many there are which give a function ^ not given by any 

 other, which function may however be a value of the 

 function given by another. And inspection of the derived 

 derangements of these groups, or consideration of the law 

 which governs their formation, enables us to determine 

 how many of the groups are to be rejected for the reason 

 that the term P does not differ in algebraic value from 

 MP, where MG is a derived derangement of the rejected 

 group G. 



