88 
[243 
243. 
ON THE THEORY OF GROUPS AS DEPENDING ON THE 
SYMBOLIC EQUATION 0“=1. Third Part. 
[From the Philosophical Magazine, vol. xviii. (1859), pp. 34—37 : Sequel to 125 and 126.] 
The following is, I believe, a complete enumeration of the groups of 8 : 
I. 1, a, a 2 , a 3 , a 4 , a 5 , a 6 , a 7 (a 8 =l). 
II. 1, a, a 2 , a 3 , /3, /3a, /3a 2 , /3a 3 (a 4 =l, /3 2 =1 , a/3 =/3a). 
III. 1, a, a 2 , a 3 , /3, /3a, /3a 2 , /3a 3 (a 4 =l, /3 2 =1 , a/3 =/8a 3 ). 
IV. 1, a, a 2 , a 3 , /8, /3a, /3a 2 , /3a 3 (a 4 = l, /3 2 = a 2 , a/3 = /3a 3 ). 
V. 1, a, /3, /3a, 7 , 7a, 7/8, 7/3a(a 2 = l, ^=1> Y 2 = L a/3 =/3a, ay = ya, £7 = 7/3). 
That the groups are really distinct is perhaps most readily seen by writing down 
the indices of the different terms of each group; these are 
I. 1, 8, 4, 8, 2, 8, 4, 8. 
II. 1, 4, 2, 4, 2, 4, 2, 4. 
III. 1, 4, 2, 4, 2, 2, 2, 2. 
IV. 1, 4, 2, 4, 4, 4, 4, 4. 
V. 1, 2, 2, 2, 2, 2, 2, 2. 
It will be presently seen why there is n<* group where the symbols a, /3 are such 
that a 4 = l, /3 2 =1, a/3 =/3a 2 . A group which presents itself for consideration is 
1, a, a 2 , a 4 , /3, /3a, /3a 2 , /3a 3 (a 4 = l, @* = 0?, a/3 =/3a);