Functions given by Groups. 317 



are equivalent. If 1342 be A, and 1423 be A~^; i = AA"^ 

 AGA"^ = G'. By the operation AG we get four substitutions, 

 abcd\ and then the four operations ^A~S <^A~^ rA~S </A~S 

 give G'. Thus we can always find G' when A and G are 

 given. But we can mostly label and identify our equivalents 

 in practice by far less cumbrous means, than such double 

 operation on the entire column of a group of high order. 



We suppose our Q equivalents to be either written out, 

 or so given that we can readily write them out, each in a 

 column or rectangle of L terms, followed by Q — i parallel 

 columns of its Q — i derivates {M.M. 7, p. 279), each derivate 

 having L terms, thus : 



Gi + lAiGi + sAiGi + 3A1G1 + + Q_i AiGi = Ni ; 



G2 + 1A2G2 + 2A2G2 + 3A2G2 + + Q-1A2G2 = N2 ] 



G3 + 1A3G3 + 2A3G3 + 3A3G3 + + Q-iAgGg = N3 ; (A,.) 



Gq + iAqGq + sAqGq + 3AqGq + + q_iAqGq = Nq. 



Here N is the group of order n ! arranged in Q different 

 ways, being exhausted by each group with its Q — i derivates. 



The derivant ;„Ag of G^ is any substitution not in G^ nor 

 in any previous derivate of Gg. It is thus impossible that 

 two derivates of any group can be alike. 



The Q maximum groups are supposed to be so given, 

 that we can readily find in our table (A„) any one of them, 

 Gy, when presented to us only by a known substitution B 

 as an equivalent of a known group G^, as G,. = 0aGr 0a ^ or 

 Gr=Q7^Gj)c- 



How this finding is to be readily done can best be 

 explained when a definite case of our problem is before us. 



5. The index-system S is 



S = a/3yyaa. . . £€f . . . 0000. . . ; (2) 



which is read, one a, one j3, two y's, two S's, three g's, &c. 

 This S, when we turn our group G into a function (3, is 

 written over unity in the group that we are placing, as we 

 say, under S, and over the first term of every derivate of G. 



