GROUPS AND MANY-VALUED FUNCTIONS. 291 



35214 45123 

 51423 51234 



The circular factor 12435 of g is obtained by multiply- 

 ing the circular factor of G by 



"^^^ =12435 = ■ 

 12345 ^^^ 



The derived of G by Q is 



QG= 12435 



24351 

 43512 



35124 

 51243 

 and the derangement of QG by Q,~^= 12435 is 



5'= 12345 

 24531 

 43152 

 35214 

 51423 

 wherefore g= 12435 Gi2435 = Q,GQ,~i. 



All that we have to do is to write^ for each one (c) of 

 the Ejv circular factors of G and its derived derangements, 

 12435(c), and for the first permutation A of each group of 

 five, 12435 A(i2435)-i= 12435 A12435, thus: 

 12345 23451 41235 34125 



2 



3 



1 



4 







4 



1 



3 



2 







3 



4 



2 



1 







5 5 5 5 

 The equivalent group 12435 G' 12435 is completed 



thus 



12345 



23415 



41235 



34125= 



= 0'4 





24531 



31542 



13524 



42513 







43152 



14253 



32451 



21354 







35214 



45321 



25143 



15432 







51423 

 The others are 



52134 



54312 



53241. 







