GROUPS AND MANY-VALUED FUNCTIONS. 317 



k 

 substitution of A elements, or of a vertical column of :g 



square groups of a substitution of B elements &c. 



These grouped groups, whose elementary groups are 

 made up of groups of A powers, or of B powers • • are 

 grouped groups of the first class. 



It is easily proved in the manner of Art. 26 that all 

 these S grouped groups are equivalent. 



38, We have restricted the auxiliary groups g that can 

 be employed in forming the groups of theorem H (37), by 

 the hypothesis of Art. (29). But in that article we saw 

 that there needs be no restriction on the auxiliary group. 



If this restriction be removed, and if circular factors 

 are admitted in the group g, made with the a elements of 

 orders Avhich do not divide A &c. in the partition N = A« + 

 &c., or made with the b elements of orders which do not 

 divide B Sec, we shall always construct grouped groups 

 of kl substitutions ; but they will not be. in a normal form, 

 that is, the principal substitutions of G will not be the 

 principal substitutions of the grouped group. 



l^or example : take 



N = g=3-3 = Aa. 

 Let G be 



123456789 



231698547 



312874965 



i?i= 123, ^2=468, ^3 = 597. 



Let the auxiliary group be 



^=123 132 

 231 213 

 312 321, 



which contains circular factors of the order 2, no divisor 

 of A. 



The grouped group constructed by five derivants Q, 

 formed by the formula in theorem H, will contain substi- 

 tutions of the sixth order, by the reasoning of Art. 30. 



