85 
But, if this limitation be removed, the constructed group 
of K l substitutions is still a grouped group. But it is not easy 
to give the exact enumeration without repetition of the con- 
structions, when g may be any group of /, made with the 
a + b + • + j elements. Every group of the order K7 so 
formed by any group g , whatever be the order of its circular 
factors, is a grouped group, whatever be the system of expo- 
nents that we employ in the denominators of Qi QjQ 3 • • • • 
(83) Grouped groups of a higher class, of which the 
elementary groups are of the order Kr, comprising r — l 
derived derangements of groups of K powers of a substitu- 
tion. We first enunciate generalisations of theorems D and G. 
Theorem J. Let 
N = A a -f B6 + Cc * * -f- Jj ; 
A 7 2; A 7 B 7 C 7 • 7 J ; b, c, • • j being any numbers, 
and K being the least common multiple of A B C • • J. 
Let X be the number of primitive roots of the congruence 
x r ^=zl (mod. M) (r 7 2), 
of which no one is comprised among the powers of another, 
M being any one of the numbers A B C • J, these X roots 
being such as are at the same time primitive or noil-priinitive 
roots of the congruences 
x r ^l (mod. X) 
where X is each one in turn of A B C • • • J. 
We can construct, R* being the number of integers (unity 
included) less than K and prime to it, 
v Att(N) 
R £ 7t a 7r b -re c • • • A" B 6 C c • • L 
different groups each of K r substitutions, all of the form 
G + 0 X G + 0 2 G + • • + 0,-! G, 
where G is a group of the order K, and 0 X 0 2 are derivants 
given in terms of one of the X primitive roots. This theorem 
J is, as I believe, a fait nouveau. 
(84.) Theorem K; 
