GROUPS AND MANY-VALUED FUNCTIONS. 333 



ways of constructing a equivalent groups J^ with the first 

 Aa elements of unity. 



By theorem K (49) we can form 



grouped groups H^ each of S^/„ substitutions with the Ka 

 elements, each containing unity. 



In like manner we can form with the BZi elements 



^'^^^^^ irbiirA.Y 

 grouped groups H^ with the next B6 elements, each of 

 8^4 substitutions j &C!. 



But further we need not choose the first Ka elements 

 of unity for the groups J^. There are 



7r(Afl)7r(N-Aa) 

 difierent ways to select the Ka elements out of N, and 

 7r(N-Aa) 

 7r(B6)7r(N-Aa-B6) 

 ways to select from N-Aa the B6 elements. 

 Wherefore there are 



-N =U 



7r(Aa)7r(B6)7r(Cc) • • 



ways to choose the Ka the BS the Cc- • elements, and we 

 have 



^ 7rN.F„F,a- ■ (M^)"(M^)^(MC)--. . ^^ 

 iraiThirC" (7rA)"(7rB)*(7rC)'^ • • 

 ways of writing under the unity of N elements a grouped 

 group H^;^ of the order S^Z^ made with Ka elements, a 

 grouped group H^ of the order 8^4 made with B6 ele- 

 ments, &c. 

 Let 



and let the grouped groups 



