m- C 408 ] 



LXV. On the Theortj of Gi-oups, as depending on the Symbolic 

 Equation ^" = 1.— Part II.* By A. Cayley, Esq.^ 



IMAGINE the symbols 

 L, M, N, . . . 



such that L being any symbol of the system, 



L-'L, L-'M, L-'N, . . 

 is the group 



1, «, A • . 



Then, in the first place, M being any other symbol of the 

 system, M~'L, M~'M, M"'N, . . will be the same group 1, a, /S . . 

 In fact, the system L, M, N . . may be written L, L«, L/3 . . ; and 

 if e. g. M = La, N = L/3, then 



M-^N = (L«)-'L/3=-.a-'L-'L;8=a-'/8, 



which belongs to the group 1, a, /3 . . 

 Next it may be shown that 



LL-\ ML"', NL-\ . . 



is a group, although not in general the same group as 1, a, ;8 . . 

 In factj writing M = La, N = Ly8, &c., the symbols just written 

 down are 



LL"', L«L-', LySL"', . . 



and we have e.g. LaL~^ . L/3L~' = Lay8L~' = L7L~\ where 7 



belongs to the group 1, a, /8. 



The system L, M, N . . may be termed a group-holding system, 

 or simply a holder ; and with reference to the two groups to 

 which it gives rise, may be said to hold on the nearer side 

 the group L"'L, L~'M, L~^N . . , and to hold on the further 

 side the group LL~\ LM"', LN~\. Suppose that these 

 groups are one and the same group \, a, ^ . . , the system 

 L, M, N . . is in this case termed a symmetrical holder, and in 

 reference to the last-mcntioued group is said to hold such group 

 symmetrically. It is evadent that the symmetrical holder 

 L, M, N . . may be expressed indifierently and at pleasure in 

 either of the two forms L, La, JjjS . . and L, aL, ^L ; i. e. we 

 may say that the group is convertible with any symbol L of the 

 holder, and that the group operating upon, or operated upon, by 

 a symbol L of the holder produces the holder. We may also 



* See January Number, p. 40. 

 t Communicated by the Author. 



