85 



But, if this limitation be removed, the constructed group 

 of K/ 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 -\-h •\- ' +7 elements. Every group of the order K/ 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 Q^ Q. Q3 • • • • 



(83) Grouped groups of a higher class^ of which the 

 elementary groups are of the order Kr, comprising r — I 

 derived derangements of groups of K powers of a substitw 

 tion. We first enunciate generalisations of theorems D and G* 



Theorem J. Let 



N = Aa -f B5 + Cc + • • -f Jy; 

 Ay 2; A y B y C y ' 7 J ; a, 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''^^l (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 non-primitive 

 roots of the congruences 



ic**^! (mod. X) 

 where X is each one in turn of A B C * • • J. 



VVe can construct, R;r. being the number of integers (unity 

 included) less than K and prime to it, 



Y X7r(N) 



R^TT air birc ' ' ' A'' B' C ' ' 3' 

 different groups each of Kr substitutions, all of the form 



G + e^G + e.G-f-'-f e,_i g, 



where G is a group of the order K, and Gj Oo are derivants 

 given in terms of one of the X primitive roots. I'his theorem 

 J is, as I believe, a fait nouveau* 

 (84.) Theorem K. 



