126] 
131 
126. 
ON THE THEORY OF GROUPS, AS DEPENDING ON THE 
SYMBOLIC EQUATION 6 n = 1.—Second Pakt. 
[From the Philosophical Magazine, vol. vn. (1854), pp. 408—409.] 
Imagine the symbols 
L, M, JS T ,... 
such that (L being any symbol of the system), 
is the group 
L~'L, L~ X M, L~ X N,... 
1, a, ß,...; 
then, in the first place, M being any other symbol of the system, M~ X L, M~ X M, 
M~ l N,... will be the same group 1, a, /3,.... In fact, the system L, M, A,... may be 
written L, La, LS...; and if e.g. M = La, N = L/3 then 
M~ X N = (La)- 1 L(3 = a~ x L~ x L(3 = a~ x /3, 
which belongs to the group 1, a, /3, .... 
Next it may be shown that 
LL~\ ML- 1 , ML- 1 ,... 
is a group, although not in general the same group as 1, a, ß,.... In fact, writing 
M—La, N=Lß, &c., the symbols just written down are 
LL- 1 , LaL~\ LßL~\... 
and we have e.g. LaL~ x . LßL~ x = LaßL~ x — LyL~ x , where 7 belongs to the group 1, a, ß. 
17—2