334 KEV. T. p. KIllKMAN ON THE THEORY OF 



be formed in any one of the Z possible ways. 



We can weave together these groups so as to form a 

 woven grouped group of S substitutions. But it is evi- 

 dent that before we weave together H^^ H^ He • we can 

 weave the a equivalent groups G^ which, with their derived 

 derangements, compose the groups J^ of S^^ substitutions. 

 Then, instead of the S^4 substitutions of 



we should have 



{H^} = {J^} + Qi{J^} + a,{J^}+ • . +Qz„_i{J^}, 

 a group of the order (S^)%. 

 Then weaving the groups 



formed in any one of Z possible ways, we obtain a woven 



grouped group of 



(S^)%(S^)%(Se)%.'.=i^ 



substitutions. 



51. We have demonstrated the theorem following — 

 Theorem L. i/" N M^ S^j F,. ly. he the numbers defined in 



Art. (49), there are constructible on the given partition ofN 



irairbirc • • irj [it A)" (ttB)* • • (ttJ^ 

 woven grouped groups each of 



substitutions. 



The largest group constructible on this partition of N is 

 evidently of 



(7rA)"(7rB)*(7rC)^- • {Tr^yirairbTTC' -irj 

 substitutions. This group is maximum; it has no derived 

 derangement; and the number of its derived groups is 

 that of its equivalents. 



For example : on the partition 



where we have 



