GIIOUPS AND MANY-VALUED FUNCTIONS. 331 



we shall obtain a group of 3 • 2 • 3, or of 3 • 2 • 6 substi- 

 tutions which cannot be arranged as a group of Artt. 

 is?} 3^)> ^^^^ ^^^ ^^ ^ grouped group of the first class. 



§ 9. 

 Woven groups : Woven grouped groups. 



49. It is known that, if 



N=A + B, 

 any group G made with A elements of the order L can be 

 interlaced with any group G' of the order L' made with B 

 other elements, so as to form a woven group of LL' sub- 

 stitutions made with N elements. 



For example : the two woven groups 





12345 6789 









23145 7689 









31245 6798 





• 





12354 7698 









23154 









31254 







can be woven into a 



group of 24 substitutions. 





For a second example, on the partition 







9 = 3-3 = A«, 







we can construct by 



aid of the auxiliary 







r=i23 









231 









312 







the grouped group 



G of A:r/=3-2'3 



substitutions, 



(48), 



123456789 



458762193 



769125438 





231589647 



584693271 



697238514 





312847956 



845971362 



976314825 





132489756 



485793162 



796138425 





321856947 



854962371 



967325814 





213547689 



548671293 



679214538, 





which is G' + RG', (26), with two derived derangements. 



G can be made into a woven group 



of the order 6^ 



thus : 



