374 KEY, T. P. KIRKMAN ON THE THEORY OF 



which is half the group of 48 = (7r2)V3, of Art. 51, made 

 with 6 = 2 '3 elements. 



This is a modular group. If we multiply it, after the 

 addition of 7 final throughout, by the seven powers of 



^=7513426, 

 we obtain a non-modular group L of 7 '6 -4 substitu- 

 tions. 



The existence of the group just written is easily proved 

 by the test, putting J' for J with 7 final, 



L = J'(i +^ + ^H • • +6') = (i + ^ + ^'+ • •6'^)J'. 



We shall return presently to the direct construction of 

 these groups of 7-6 = 4, which are of considerable interest, 

 by reason of the learned and instructive researches of 

 Galois, who first af&rmed their existence, as well as that 

 of the groups of 1 1 • 1 o • 6 made with eleven elements {vide 

 Hermite's Theorie des Equations modidaires, § xiv.), and 

 also by those of MM. Betti of Pisa, Kronecker of Berlin, 

 and Hermite of Paris, who appear all to have constructed 

 these groups. 



Has anything more been done during the last twenty 

 years in this theory of groups, apart from the recent com- 

 petition, beyond the construction of these groups implicitly 

 discovered by Galois, and Mr, Cay ley's analysis of groups 

 of the eighth order ? 



It suffices for my purpose here to shew, that a vast 

 number of non-modular groups, and consequently often 

 of modular groups made by adding to them their derived 

 derangements, is given by the theorems demonstrated in 

 the preceding Memoir. 



85. To form the modular group of 8 •7-6 -4, we add 

 8 final to every substitution of L, and then multiply L' 

 thus formed by the group F 



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