90 



Neither the group of 18 nor that of 36 can be presented as 

 a grouped group of K/ substitutions of theorem H, like 

 the following of K/ = 6 • 3 substitutions. 

 123 456 789 789 123 456 789 456 123 

 231564897 897231564 897 5 64 231 

 312645978 978312646 978312645 

 45v3 789 12 3 123 789 456 456 123 789 

 564897231 231897564 564231897 

 645978312 31297864 5 645312978 



Woven Groups. 



(95) " Let N - A + B. 



Let G be any group of L substitutions, formed with A 

 elements, and G\ be any group of L^ substitutions formed with 

 B elements following the A in unity. We can always form 

 a woven group of LL^ substitutions. There is nothing to 

 ]>revent (5 and G^ being themselves woven groups. 



€. g* the two woven groups : — 



12 3 4 5 and 6 7 8 9 

 2 3 1 4 5 7 6 8 9 

 31245 6798 

 12354 7698 



2 3 15 4 



3 12 5 4 



will form a woven group of 24 substitutions." 



Woven Grouped Groups. 



(96) The grouped group J+Qi J+QJ of (89) of 3'2'3= 

 hrl substitutions can become a woven grouped group of 

 6^'3=:648 substitutions. For J can be woven into a group 

 of G^ thus : 



123456789 123546789 



123456897 123546897 



123456978 123546978 



123456879 123546879 



123456987 123546987 



123456798 123546798 &c, 



