GROUPS AND MANY-VALUED FUNCTIONS. 3^7 



groups g^.; that i%, if 



^1=1, A,A^..A^-^ 



we have 



Let the N - 1 groups gig<i' • be written^ and let each 

 group have written under it its N - 2 didyraous radicals, 

 viz., 



«iC!2^3- •ajv-2 under g^, 

 b^bj)^ • • 6jv_3 under g^, &c. 



We have a system of N - 1 groups hjiji^ • • of the order 

 2(N-2) in each of which, h^, is a didymous factor of 0, 

 viz. Tq, which is permutable with each of the N - 2 didy- 

 mous radicals of h^. 



79. Every group of m powers 



when m is even, has one power <p^"^ permutable ivith each 

 of the m didymous radicals of the group. 



It is easily seen that if h^ be any one of the N - 1 

 groups h-jiji^' • 



are the remaining N - 2 groups. That is, if 7 be any one 

 of the (N-i)(N-2) didymous radicals in the groups A^Aa* • 

 and h be any other, we have always 

 7=6'«86'-", or 



for some value of a. 



The didymous radicals in the group h^ are alternately 

 of the two forms, (Art. 27), 



N-i = 2- — ~ + i''i =Aa + B6 

 2 ^ 



N-2 



]S[_i=2. l-i'i, =A« + B5, 



and those in the group h^ have all the element r undis- 

 turbed. 



Let us denote by H the entire system of (N- i)(N-2) 



