79 
Def. If PG and GP are the same group of permutations, 
PG is a derived derangement of G. 
(8) Theo. A. The derived groups of G (including G), 
lit 
are derangements of the groups G GjG 3 • • G x equivalent to 
G ; and there are among these derived groups neither more 
nor fewer derangements of G than of any group equivalent 
to G. 
(9) Cor. If the number of groups equivalent to G is 
7 r 
N 
k 
G has has no derived derangement, except itself; and 
every derangement of G is a group of permutations different 
from every derived of G. The same thing is true of the 
derangements and derived groups of every equivalent of G. 
If the number of groups equivalent to G be fewer than that 
of its derived groups, we know that G lias derived derange- 
ments. 
If M be the number of groups equivalent to G, is the 
number of derived derangements of each (G 1 ) of these equiva- 
lents (including (G 1 ) in this number), which are found among 
G and its derived groups. 
Each of the equivalent groups G G, . . . G L forms with its 
derived derangements a different group (containing (1)) of 
substitutions. 
M 
I believe that this theorem A (8) and the corollary (9) that 
follow it, are among the /aits noaveaux of my Memoir. 
They lead directly to all the enumerations and extensions 
following. 
(10) Theorem B. The number of different groups of the 
order N (1, A, A 2 , . . Af r_1 ) made with N elements, which 
are N powers of one substitution, is ■ 7r being the 
number of integers less than N and prime to it, unity 
included. 
