(1) denotes the natural order of the elements, is the sub- 
stitution ~ •” 
“ Def. A system of R arrangements of N elements 
(1) A x A 2 • * A*_j _ 
is a group of k substitutions. 
0) A • . • 
U)’ (1)’ (1)’ 
if both the products A A„ and A„ A,„ are substitutions of the 
group, A„, A„ being any pair of the group.” 
“ Def. Let G be any group of h substitutions made with 
N elements not containing the substitution P. The product 
GP = (1 +A 1 -l-A 2 + ,, -t- A*_i) P. 
is a vertical column of k permutations — 
P 
A X P 
A 2 P 
• 
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 r „P, A being any substitution of G.” 
G has different derangements; (jrN=l 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 
7rN 
T 
different derived groups (PG.) 
*Let 
p-i 11 } 
i - p 
Def. The groups G and PGP' 1 are equivalent groups 
containing (1), if they be not the same group (1 A, A,* • A a _i). 
