Functions given by Groups. 323 



Or thus : let the reader write over i 2 3 . . ;^ any S he 

 pleases ; under that unity let him write any permutation 0,- 

 he pleases made within the clusters, so as to exchange no 

 two elements not in the same cluster, and therefore not 

 carrying the same index. Let him now write any permuta- 

 tion whatever of the n elements, calling it H ; and next let 

 him perform, without regard to indices, the tactical operation 

 H0i upon 0i, and call the result K. If, now, he writes over 

 hoth H and K his selected S, he will find them algebraically 

 identical. 



Hence, it is clear, that if every substitution of G^ in turn 

 be taken for H, we have it demonstrated that 



Oa = Q>A\ (9) 



whichever among the*/— m 0's of I^+i 0, may be. 



10. The two sides of (9) are the same rectangle of L 

 products of powers. No man, therefore, will deny that, if 

 both sides be multiplied by unity, their algebraic identity 

 will remain visible. Let us multiply (9) on left and right 

 by 



We get 



6d-Oi(0r'6d0O = 0.(5e, (10) 



whichever of the O's of I^+i 0^ may be ; where G, is a group 

 to be readily found in (A)„, art. 4, by the known 0^ and the 

 known G^, in the definition 



Ge = 0r^GA, (11) 



and (10) reads, G^ under 2 is algebraically identical with 

 the derivate fl^G^ under 2 of the given group G^. 



That Qfj^ is a derivate of G^, follows from the truth that, 

 as Qi is not in G^ (for if it were (11) would give 



0A0r'=,Ge = G^, Q.E.A.) 



but is in Ng in (A„), it must occur in some derivate of Gg ; 

 and that derivate, whether 0^ be or be not the first of its L 

 substitutions, is the derivate Qfi^ of Ge {M.M. p. 280). 



