332 



REV. T. P. KIRKMAN ON THE THEORY OF 



123456789 231586749 312846759 



123459687 231589647 312849657 



123457986 231587946 312847956 



123459786 231589746 312849756 



123456987 231586947 312846957 



123457689 231587649 312847659, &C, 

 Let this group of the order 6^ be (G) ; then 

 (0)11+458762193 + 769125438} 

 is a woven grouped group of the order 648. 

 50. Take the partition 



N=Aa + B6 + Cc+ . . +JJ 

 A^B, B^C, C^D, &c. 

 Let Mb be the number of groups equivalent to any 

 group G^ made with R elements, of S^j substitutions, and 

 such that G^ shall not be equivalent to G^, even though 



A=:B. 



The model group G^ is determined by a certain partition 

 E = HiVi + U^r^ + ^3^3 + • • 

 and by a certain system of circular factors, and may be 

 any group constructible with R elements on this partition. 

 Let F„ be the entire number of model groups of the 

 order /„ constructible with a elements. 

 We can form of the order S^/ 



Mj equivalent model groups J^^ 

 with Aa elements, by selecting any one of the M^ groups 

 G4 to be made with the first A elements 123- -A, any 

 other to be made with the next A elements of unity, &c. 

 But this order of the elements is not necessary. We may 

 choose the A elements a times out of Aa in 



77 ( Aa) 



different ways. Wherefore there are, {'ira= 1 • 2 • 3 • • a), 

 7r(Aa) 



7rflt(7rA) 



«(M. 



