372 REV. T. p. KIRKMAN ON THE THEORY OF 



theorem, that if to a group G you add any number of 

 derived derangements of G, 



PG + QG + RG &c. = GP + QQ + QTI+ • -, 

 such that every power and product of the derivants PQR • ' 

 is found in the series PQE, • • » the sum M of substitutions so 

 obtained is a group. 



This principle has remained since the days of Cauchy 

 little more than a broad and barren generality. It is not 

 an easy step from this proposition to the answer of the 

 following questions ; 



I. How many derived derangements at the most can be 

 added to the model group G ? 

 ' II. How many different inferior modular groups can be 

 selected from the maximum modular group ? 



III. How are we to form these modular groups, both 

 maximum and subordinate? 



IV. What is the number in each case of the possible 



equivalent groups ? 



V. How are we to determine, when a non- modular 



group, which is not simply a group of powers of ^, 

 is found, the number and order of modular groups 

 of which it is the model ? 

 I am not aware that these questions have been discussed 

 elsewhere than in this Memoir. 



The first and the last of these questions are, if I mis- 

 take not, sufficiently answered by my theorems, including 

 the corollary to theorem A, (9) ; and this corollary, along 

 with the theorem H!, (37), is an instrument of new and 

 considerable power, as we shall presently see. 



84. As an example, which will introduce certain import- 

 ant groups that are known, and a superior group which is 

 probably new, let us take the partition 

 8 = 2-4=A-A^ = Aa. 

 The theorem H, (37), gives thirty for the number of 

 equivalent grouped groups (F) of eight substitutions^ all 



