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



equivalent modular groups of 



= 8-7-3 



240 ' -^ 



substitutions, made by adding to a group of fifty-six two 

 derived derangements. 



This group of 8 • 7 • 3, like that of 8 • 7, is subordinate to 

 the maximum modular group V of 8 "7 -6 -4, and forms 

 part of it. V can be "written either as the group F with 

 twenty derived derangements, or the group of fifty-six 

 with two. 



The group of 8 • 7 is one of the groups included under 

 M. Mathieu^s theorem, that whenever N is a power of a 

 prime number, there is a group of N'(N-i) substitutions. 



All these groups of N'(N-i) are modular, and form 

 subordinate portions of the maximum groups given by the 

 theorems H (37) and A (9). 



The theory of them all is like that of the above groups 

 of 2^- (2^- 1) ; and the construction of these groups on any 

 partition 



n being a prime number, is easy, as well as the enu- 

 meration of their equivalents, which, being known, the 

 theorem A gives the maximum groups to which they 

 belong. 



87. Non-modular groups of N-N-i N-2 substitutions 

 when N - 1 =^', p being any prime number. 



These groups were first discovered by M. Mathieu ; but 

 nothwithstanding the elegance of his demonstration, by 

 the method of imaginary subindices, there are few readers 

 who will not desire a mode of investigation somewhat less 

 learned. The following is a sketch of a tactical demon- 

 stration of these groups, and is little else than a repetition 

 of that given above for i=i. 



The group (L) of N - 1 substitutions, which is formed 

 by the theorem H, (37), on the partition 



