GROUPS AND MANY-VALUED FUNCTIONS. 279 



neither in G nor in GP. Then no arrangement in GQ 

 can be G ; nor can it be in GP ; for if 



we should have 



that is^ Q is in GP, contrary to hypothesis. 



Hence we see that G, GP, and GQ are what Betti has 

 called equal groups, that is, groups of permutations, all 

 whose substitutions are identical, the common model 

 being G. 



In this way we can partition the entire system of 

 1 • 2 • 3 . . N permutations of N elements into 

 i'2-3- -N 

 ~ k 

 equal groups, by adding to 



G, GP, GQ. . 

 the derangement GR of G by R, which is not in G-f GP 

 + GQ. 



Hence we see that G has -^~- - i derangements, and 



a; 



that it has no more, and that these are all equal groups. 



y. Let G be a model group of the order k, made with 

 N elements, viz,, 



G=i+Ai + A2+..+A;t_ij 

 and let P be any substitution not in G. 



The product 



PG=:P + PAi + PA2+ . . +PA;,_i 



is a group of substitutions. In fact it is the derangement 

 by P of the model group 



PGP-^= 1 + PAjP-i + PA2P-1 + . . , 

 which is a group, because, according to the definition (2)^ 



PA^P PA,iP =PA,„A^1 =PAgP . 

 The group PQ is the derivate of G by P. 

 No arrangement of PG is iu G ; for if 

 A —PA 

 we should have • 



