214 THE REV. T. P. KIRKMAN ON NON-MODULAR GROUPS. 



it is perhaps wortli while to have spent so much time on 

 the method_, for the sake of showing that we can dispense 

 in the construction of these groups with congruences_, and 

 with the imaginary subindices so ably handled by MM. 

 Betti and Mathieu. Besides this^ it may not be useless to 

 show the connexion between the theory of groups and that 

 of combinations ; and the theorem that we have proved, 

 that the duads made with 25 elements can be exhausted in ten 

 6-pIets, six ^-plets, and fifteen \-plets,i^ of itself deserving of 

 record and of an example. 



The theory of combinations appears to me, with my pre- 

 sent light, to be likely to owe more than it can contribute 

 to that of groups. The theorem about the duads of 25 is 

 obtained by the study of the group of 120 made with five, or 

 of M. Mathieu^s group of 120 made with six elements. It 

 is a simple case of this more general proposition (proved by 

 inspection of the groups of his general theorem), that groups 

 of (N+ i)N . (N— i) can always be formed with N + i ele- 

 ments when N is prime : 



Theorem : When N is any prime number, N* elements can he 

 thrown into ^N . (N + 1) (N — i) (N — i-plets), N + i N-joZe/5, 

 and ^N . (N— i) (N + i)-plets, so as to exhaust once and once 

 only the duads possible with the N^ elements, 



I shall content myself with giving the twenty-eight 

 6-plets, the eight 7-plets, and the twenty-one 8-plets, 

 which can thus be made with the 28 + 21 =49 elements, 

 abed efg hijklmnpqrstuvwxyzu^fyj 

 ABCDEFGHIJKLMNPaHSTUV. 

 bOcM^ ^Q^B^K xYd^ul, tVhmCt 

 oDlXkE yC^SiJj elmJwV i¥kQuT 

 aGmBnY wTbVaB yTeGpK IGjYsS 

 aUqCpl yVbYy'E fUnJjyV jBmLxR 

 fJsArK rr/3PcB qFfYccM ffllcSrV 

 ehuAtM cNifG^U gEpMzU vJnEiN 

 hMcBvS vTdUsR hDqKsl^ jJaCzQ. 



