Functions formed on Groups. 39 



Two, (3i ' ' and Oik " ' each of 6 terms and 60 values. 

 One (Bud* '1 of 12 terms and 15 values. 

 Thirteen others, <3£> ' ' above written as G rD , each of 1 2 

 terms and 60 values. 



S = a/3yyyy. 



8. We consider again the two groups G rf = 356142, and 

 G 2(i = 542631 of art. 6. Both have 9=124365 in common 

 with our new index group I (+ , = I 24 . 



But we cannot write L 4 as the product of two groups 

 H l2 , and J 3 = 1 + 124365, because the only H 12 contains this 

 Jn. We must then write l 2i as the product of three 

 groups, thus : 



I 24 = H 6 -K 2 -J 2 = 123456 x 123456 x 123456 =H fi 1R4 

 \1i24536 125634 = * 124365 = 6 

 X 2 i25346 

 * 3 i23546 

 X 4 i 25436 

 \Bi24356 



We require here another theorem. 



Theorem U. If l m is any group of n elements, of 

 order t+i, and is the product of two groups H A+1 and 

 IKuH-iXm+D °f which the latter is the product of two groups 

 K k+1 and J m+1 , such that no two of the groups H, K, and J 

 have a common substitution, and where 



H ft+ i=i +\i + X 2 + .. +\ h , 



K.jfc+1 = I + Tjl + ?/ 2 + . . + 1] k , 



Jm+i = i + ei + e 2 + ,, + e„ 1I 



so that, in form more explicit than in (F.G. 8, 7). 



I t +i = 1 + Qi + ©2 + . . +Q m 

 + \i + \ 2 + . . +\ h 



+ V1 + V2+ " + >lk 



+ d x + 2 + • • + Qt-m-h-k ) 



then, if G d be any maximum group of the n elements, 



