GROUPS AND MANY-VALUED FUNCTIONS. 305 



such that the first of the diflFerences 



A -A', a- a', B-B', h-h', C-C, c-d ■■, 

 which is not zero^ is positive. If they are all zero, Q, also 

 is a principal substitution. 



We are about to enumerate groups having their prin- 

 cipal substitutions defined by this partition of N. 



A grouped group has elementary groups of which those 

 of the same form are permuted, sometimes as though they 

 were single unchangeable elements, and more frequently 

 so that they suffer certain interior derangements while 

 they are transposed. For example, 



1234 



2143 



3412 



4321 

 is a grouped group, in which are seen two elementary 



groups and •^^• 

 ^ ^ 21 43 



Another grouped group is 



1234 

 2143 

 3421 

 4312. 

 In the first the elementary groups are simply permuted ; 

 in the second they are permuted while one suffers at the 

 same time a certain interior derangement. 



28. Let G be a group of theorem C (12) ; and let 



PlP2"Pa 



be the a circular factors of A*'^ order, 



9a+l ^a+2 ' ' ^a+b 



the b of the B"' order, 



the c of the C* order, and let 



P\n=Pm^ 



be the «*'' cyclical permutation of /i^j, &c. 



SER. III. VOL. I, RR 



