GROUPS AND MANY-VALUED FUNCTIONS, 281 



G=P^GP-^ 

 Let P'"=i, we have 



G(i +P + P'+ • • +P™-^) = (i +P + P'+ • • +P'"-')G. 

 or G^=^G = G', 



g being the group of the powers of P ; and G' is a model 

 group of the l^mP^ order ; for 



A^ r A„ r = A,„ r P Ap = A^ "' A^ = A^ A^ r = A^ i: , 

 which is the test of a group by the definition, (2) . 



We shall call the derangements GP, GQ, GR • • of Gj 

 which are also the derivates PG, QG, RG- -of G, the 

 derived derangements of G. And we have the known 

 theorem. 



If PG=GP he a derived derangement of G, G being of 

 the order k, and P being a substitution of the m*'* order, 

 there is a group of the km}'^ order composed of G and its 

 derived derangements by the powers of P. 



It is also known that if PG, EG, QG • • be derived de- 

 rangements of G, the group G with its derangements by the 

 powers and by the products of the powers of PQK • • forms 

 a, model group of substitutions. 



9. Let us suppose that of the equivalent or identical 

 groups above mentioned 



PGP-i and QGQ-^ 

 are identical, and diflPerent from G. 



From 



PGP-i=QGQ-i 



it follows that 





Q-iPGP-^Q=G, 



Let 





Qr^V=e, 



whence 







which gives by 



what precedes 







eGe-'=G, 



SER, III, VOL. 



I. 



©o 



