78 



(I) denotes the natural order of the elements, is the sub- 

 stitution — ' ." 



" Def. A system of R arrangements of N elements 

 (1) Ai A2 • • • A,_i 

 is a group of k substitutions. 



(1) Ai 42 . . . 

 (1)' (1)' (1)' 



if both the products A A„ and A« A„» are substitutions of the 

 group, A,„ An being any pair of the group." 



" Def. Let G be any group of k substitutions made with 

 N elements not containing the substitution P. The product 

 GP = (1 + Ai + A2 + • • + A,.0 P. 



is a vertical column of k permutations — 



P 

 AiP 

 A2P 



which differs from G written in a vertical column only in the 

 horizontal order of entire vertical rows of elements, 

 GP is the derangement of G by P." 



Theo. ''The derangement (GP) of G by P is the 

 derangement of Gby A,„P, A being any substitution of G.*' 



G has ^^ different derangements; (ttN^I 2 3 • • N). 



Def. '' The product PG of the same P and G is the 

 derived of G by P" 



Theo. " The derived (P G) of G by P is the derived of 

 G by (P A,.)." 



G has ^different derived groups (PG.) 



Let P-^=:^^ ; 



Def. The groups G and PGP"^ are equivalent groups 

 containing (I), if they be not the same group (1 Ai Ao • * A^-i), 



